Symbolic Computation of Conserved Densities
for
Systems of Nonlinear Evolution Equations
by
Ünal GÖKTAŞ
ii
A thesis submitted to the Faculty and the Board of Trustees of the Colorado
School of Mines in partial fulfillment of the requirements for the degree of
Master of Science (Mathematical and Computer Sciences).
Golden, Colorado
Date
Signed:
Ünal Göktaş
Approved:
Dr. Willy A. Hereman
Thesis Advisor
Golden, Colorado
Date
Dr. Graeme Fairweather
Head of Department
Mathematical and Computer Sciences
æ
ABSTRACT
An algorithm for the symbolic computation of polynomial-type conserved densities for systems
of nonlinear evolution equations is presented. The algorithm is implemented in Mathematica.
The code CONDENS.M is tested on many systems involving well-known equations from
soliton theory. CONDENS.M automatically carries out the lengthy symbolic computations
for the construction of conserved densities and associated fluxes. For systems with parameters,
CONDENS.M can be used to determine the conditions on the parameters so that a large
number of conservation laws will exist. The existence of a sequence of conserved densities is a
predictor for integrability of the system.
Abstract
æ
iv
CONTENTS
Dedicationx
1. INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2. COMPUTATION OF CONSERVED DENSITIES . . . . . . . . .
2.1 Euler-Lagrange Equations . . . . . . . . . . . . . . . . . . .
2.2 Conservation Laws . . . . . . . . . . . . . . . . . . . . . . .
2.3 Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3.1 Determining the Weights of Variables and Parameters
2.3.2 Constructing the Form of the Density . . . . . . . . .
2.3.3 Determining the Unknown Coefficients . . . . . . . .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
3. APPLICATIONS AND EXAMPLES . . . . . . . . . . . . . . . . . . . . .
3.1 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.1.1 Fifth-Order Korteweg-de Vries Equation . . . . . . . . . . . .
3.1.2 Seventh-Order Korteweg-de Vries Equation . . . . . . . . . . .
3.1.3 Hirota-Satsuma System . . . . . . . . . . . . . . . . . . . . . .
3.1.4 Testing Numerical Algorithms for PDEs . . . . . . . . . . . .
3.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2.1 Korteweg-de Vries (KdV) Equation . . . . . . . . . . . . . . .
3.2.2 Modified Korteweg-de Vries (MKdV) Equation . . . . . . . . .
3.2.3 Generalized Schamel (GS) Equation . . . . . . . . . . . . . . .
3.2.4 Ito System . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2.5 Nonlinear Schrödinger (NLS) Equation . . . . . . . . . . . . .
3.2.6 Derivative Nonlinear Schrödinger (DNLS) Equation . . . . . .
3.2.7 Modified Vector Derivative Nonlinear Schrödinger (MVDNLS)
3.2.8 Kaup System . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2.9 Dispersiveless Long Wave System . . . . . . . . . . . . . . . .
3.2.10 Broer-Kaup System . . . . . . . . . . . . . . . . . . . . . . . .
3.2.11 Drinfel’d-Sokolov System . . . . . . . . . . . . . . . . . . . . .
3.2.12 Boussinesq Equation . . . . . . . . . . . . . . . . . . . . . . .
3.2.13 3-Component Korteweg-de Vries Equation . . . . . . . . . . .
3.2.14 2-Component Nonlinear Schrödinger Equation . . . . . . . . .
.
.
.
.
.
.
.
.
.
.
.
.
.
. . . . .
. . . . .
. . . . .
. . . . .
. . . . .
. . . . .
. . . . .
. . . . .
. . . . .
. . . . .
. . . . .
. . . . .
. . . . .
Equation
. . . . .
. . . . .
. . . . .
. . . . .
. . . . .
. . . . .
. . . . .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
1
4
4
10
11
12
16
19
26
26
26
30
34
35
35
35
36
37
38
38
39
39
41
41
42
42
43
44
44
4. USING THE PROGRAM CONDENS.M . . . . . . . . . . . . . . . . . . . . . . . . . 46
Contents
5. OTHER SOFTWARE PACKAGES
5.1 Existing Software Packages . .
5.1.1 SYMCD . . . . . . . .
5.1.2 DELiA . . . . . . . . .
5.1.3 FS . . . . . . . . . . .
5.2 Software under Development .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
vi
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
53
53
53
54
56
57
6. CONCLUSION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
LIST OF FIGURES
LIST OF TABLES
3.1
3.2
3.3
Conserved Densities for Sawada-Kotera and Lax 5th-Order Equations . . . . . . 29
Conserved Densities for Kaup-Kupershmidt and Ito 5th-Order Equations . . . . 31
Conserved Densities for Sawada-Kotera-Ito and Lax 7th-Order Equations . . . . 32
List of Tables
ix
Acknowledgements
I would like to express my appreciation and thanks to Dr. Willy Hereman, my thesis advisor,
for his helpful suggestions and great assistance in making this project possible.
Many thanks go to my thesis committee members, Dr. Steven Pruess and Dr. John DeSanto,
for their comments and suggestions. Special thanks goes to Dr. Frank Verheest (University of
Ghent, Belgium) for his invaluable comments.
I greatly appreciate the generous financial aid from the Turkish Government and the Department of Mathematical and Computer Sciences, which both supported my Master’s studies. My
research was also supported in part by the National Science Foundation under Grant CCR9300978.
Finally, my deepest and greatest gratitude goes to my parents for their encouragement and
support.
æ
DEDICATION
To my parents and my brother
æ
1. INTRODUCTION
The discovery of the soliton, its remarkable properties, and the incredible richness of structure
involved in its mathematical description occurred in two stages and over a period of almost
140 years. The story begins with the observation by John Scott Russell of “the great wave
of translation”, while riding on horseback beside the narrow Union Canal near Edinburgh,
Scotland, in 1834 [1, 32]. After watching a pair of horses pull a barge along the canal, Russell
wrote that when the boat stopped, water “rolled forward with great velocity assuming the
form of a large solitary elevation, a rounded heap of water, which continued its course without
change of form or diminution of speed.”
Subsequently, Russell, a civil engineer, did extensive experiments in a laboratory scale wave
tank in order to study this phenomenon more carefully. From watching wave after wave in the
tank, Russell confirmed that solitary waves exist.
When Russell reported his findings, many leading scientists were incredulous. In 1849, however,
Stokes, an Irish physicist, supported Russell’s claims by showing that a solitary wave could arise
from a combination of periodic waves [8]. Nevertheless, a precise mathematical explanation of
the waves remained unknown for more than half a century after Russell’s first observation.
The mathematics describing solitary waves took a step forward in the 1870s. Boussinesq, a
French mathematical physicist, and Rayleigh, an English physicist, independently produced
descriptions of a solitary wave’s profile by using the basic equations of fluid dynamics [8].
Do the equations for water waves allow for the existence of solitary-wave solutions? This
question was finally resolved in 1895, when the Dutch physicist Diederik Johannes Korteweg
and his student Gustav de Vries derived an equation that supported the existence of solitary
waves, which now bears their names.
Despite this early derivation of the Korteweg-de Vries (KdV) equation, it was not until 1960
that any new application of the equation was discovered.
In 1955, the physicists Fermi, Pasta and Ulam studied a one-dimensional anharmonic lattice
of equal masses coupled by nonlinear strings. They felt that any smooth initial state would
eventually relax into equilibrium due to the nonlinear couplings. Much to their surprise, rather
than equilibrating, the energy eventually recurred [1].
This discovery remained largely a mystery until Zabusky and Kruskal started to investigate
the system again in 1965. From a detailed numerical study they found that stable pulse-like
waves could exist in a system described by the KdV equation. A remarkable property of these
solitary waves was that they could collide with each other and yet preserve their shapes and
speeds after the collision. Accordingly, Zabusky and Kruskal named them solitons, following
the practice in physics of giving elementary particles names that end in on, such as proton,
electron, etc. The first success of the soliton concept was that, indeed, it helped to explain the
recurrence in the Fermi-Pasta-Ulam system [27].
The discovery by Zabusky and Kruskal created renewed interest in the equations for solitary
1. INTRODUCTION
2
waves and the special properties of their solutions. New more powerful methods for describing
the waves mathematically have been developed. Many equations have been found to have solitary waves as solutions. Moreover, solitary waves have been observed in a variety of natural
realms: in the atmosphere, in oceans, in plasmas, and possibly in nervous systems of living
organisms. Finally, solitons started playing an important technological role in modern telecommunications. Their persistent shape and immunity to distortion make them suitable carriers
of long-distance signals [8].
The next challenge was the analytical understanding of the numerical results. One of the earliest
indications that certain partial differential equations (PDEs) of evolution type might have
special properties arose with the discovery of an infinite number of independent conservation
laws for the KdV equation [47]. Two obvious conservation laws, corresponding to conservation
of momentum and energy, were already known, and Whitham [43] had found a third, namely
the conserved density that corresponds to Boussinesq’s famous moment of instability. Zabusky
and Kruskal searched for more conserved densities and discovered a fourth and fifth. Their
method indicated that the equations for the coefficients at stage six were overdetermined (as
they are thereafter) and, therefore, they were not particularly surprised when they did not find
a conservation law at level six. However, they had made an algebraic mistake, and this caused
a delay of more than a year before they continued on the right track [32].
The next surge of momentum came with the arrival of Miura who was asked by Kruskal to get
his feet wet by searching for a conservation law at level seven. He found one and also filled in
the missing sixth. Eight and nine fell quickly and Kruskal and Miura were fairly certain that
there was an infinite number. However, rumors originating from the Courant Institute that
nine was the limit challenged Miura to find the tenth. He did so during a two-week vacation in
Canada in the summer of 1966. There is also a rumor that he was seen about that time in Mt.
Sinai, carrying all ten. It was now clear that there was a conservation law at each level [32].
The existence of an infinite sequence of conservation laws was an important link in the chain of
discoveries. For example, it led to a one-parameter family of Bäcklund transformations, named
after Miura and Gardner, that connect the solutions of KdV and the modified KdV equations.
From this the famous Lax pair was found, which associates a couple of linear equations to the
KdV equation. From that, the inverse scattering technique (IST) for direct linearization of
integrable equations was developed. It was also discovered that the KdV equation, and many
others like it, can be written in bi-Hamiltonian form [47].
These developments covered most of the properties of integrable systems. A comprehensive
definition of the term integrable is proving to be elusive. Integrable systems are in some sense
exactly solvable and exhibit globally regular solutions for all initial conditions. In contrast, the
term nonintegrable is, generally, taken to imply that a system can not be solved exactly and
that its solutions can behave in an irregular fashion due to sensitivity to initial conditions [47].
Given the existence of an infinite sequence of conservation laws, most of them without physical
interpretation, one might ask “so what?” Besides some theoretical implications for the description of solution manifolds, of what practical use are these conservation laws? The answer is
that they provide a simple and efficient method to study both quantitative and qualitative
properties of solutions [40]. There seems to be a close relationship between the existence of
a sequence of conservation laws and integrability. However, the nonexistence of an infinite
number of conservation laws does not preclude integrability. There are indeed equations which
only have a finite number of conservation laws, which are not Hamiltonian (to the contrary,
1. INTRODUCTION
3
they are dissipative!), and which nevertheless can be directly integrated. The most notable of
these is the Burgers equation. It can be integrated by directly transforming it into the linear
heat equation with the Cole-Hopf transformation [47].
In this thesis, we present an algorithm for computing polynomial-type conservation laws for
systems of nonlinear evolution equations. We also present a symbolic software package CONDENS.M (in Mathematica) which automates the tedious algebra and differential calculus
needed in the explicit computations.
In Chapter 2, we describe the algorithm for the computation of conserved densities for systems
of nonlinear evolution equations. Since the Euler-Lagrange equations from calculus of variations
play an important role in the algorithm, a brief introduction on that subject is included.
The algorithm is implemented in Mathematica, and tested on well-known evolution systems of
soliton theory. In Chapter 3, we give some of these results. Also, we address applications of
our program CONDENS.M. In particular, we show how it can be used to search integrable
systems. In a search of integrable fifth-order evolution equations, we retrieved all the known
integrable cases. In a similar search of seventh-order evolution equations, apart from the known
cases, we found one case that is not mentioned in the literature.
In Chapter 4, we describe the usage of our program. Also, its capabilities and limitations are
indicated.
In Chapter 5, a comparison with other existing programs is given. Moreover, continuing projects
are addressed.
We offer the scientific community a symbolic package to carry out the tedious calculations of
conserved densities for systems of nonlinear evolution equations. The software with data and
output files is available from different sources.
æ
2. COMPUTATION OF CONSERVED DENSITIES
In this chapter we describe our algorithm to compute conserved densities of systems of evolution
equations. Since the Euler-Lagrange equations from calculus of variations play an important
role in our algorithm, we start with a brief introduction about these equations and explain how
we use them.
2.1 Euler-Lagrange Equations
We first give the basic lemma that we will use in deriving the Euler-Lagrange equations.
Lemma 2.1: If x1 andZ x2 (> x1 ) are fixed constants and G(x) is a continuous function for
x1 ≤ x ≤ x2 , and if
x2
η(x)G(x) dx = 0 for every choice of the continuously differentiable
x1
function η(x) for which η(x1 ) = η(x2 ) = 0, then G(x) ≡ 0 in x1 ≤ x ≤ x2 .
Proof: Proof can be found in [42, pp. 16].
Now we proceed with the derivation of the Euler-Lagrange equations. Calculus of variations is
concerned with the existence and determination of functions that extremize a definite integral.
Such functions must satisfy differential equations, called the Euler-Lagrange equations.
Theorem 2.1: The differential equation that has to be satisfied by the continuously differentiable
function that extremizes the integral
I=
Z
x2
f (x, y, y 0 ) dx,
(2.1)
x1
with the prescribed conditions y(x1 ) = y1 , y(x2 ) = y2 , is given as
∂f
d ∂f
− ( 0 ) = 0.
∂y dx ∂y
Proof: We denote the function that extremizes (2.1) by y(x) and form the one-parameter
family of “comparison” functions Y (x), defined by
Y (x) = y(x) + η(x),
(2.2)
where η(x) is an arbitrary differentiable function for which η(x1 ) = η(x2 ) = 0 and is the
parameter of the family. The importance of the form (2.2) is that y(x) is the member of the
family for all the choices of η(x) when = 0.
2. COMPUTATION OF CONSERVED DENSITIES
5
Geometrically, we are dealing with the family of curves connecting the points (x1 , y1 ) and
(x2 , y2 ). The extremizing (in fact minimizing) arc y(x) is a member of each family for = 0.
Replacing y and y 0 in (2.1) by Y (x) and Y 0 (x), respectively, we obtain the integral
I() =
Z
x2
f (x, Y, Y 0 ) dx,
(2.3)
x1
where for a given function η(x), this integral is clearly a function of the parameter . From
(2.2), we have
Y 0 = Y 0 (x) = y 0 (x) + η 0 (x).
(2.4)
Also from (2.2), we see that setting equal to zero is equivalent to replacing Y and Y 0 by y and
y 0 , respectively. So the integral (2.3) will be a minimum with respect to when = 0. This
fact holds for any choice of η(x). Thus, the problem is reduced to an ordinary minimization
problem of calculus with respect to the single variable . Moreover, we know the minimum is
achieved at = 0, which gives us that I 0 (0) = 0. Differentiating (2.3), we get:
dI
= I 0 ()
d
∂f ∂Y 0
∂f ∂Y
+
) dx
∂Y 0 ∂
x1 ∂Y ∂
Z x2
∂f
∂f 0
=
(
η+
η ) dx.
∂Y 0
x1 ∂Y
=
x2
Z
(
(2.5)
Extremization requires I 0 (0) = 0, and therefore,
I 0 (0) =
Z
x2
x1
(
∂f
∂f
η + 0 η 0 ) dx = 0.
∂y
∂y
Integrating the second term in the integral by parts, we obtain:
∂f x2 Z x2
η +
I (0) =
∂y 0 x1
x1
Z x2
∂f
= 0+
−
∂y
x1
= 0.
0
!
∂f
d ∂f
− ( 0 ) η dx
∂y dx ∂y
!
d ∂f
(
) ηdx
dx ∂y 0
(2.6)
Since (2.6) must hold for all η, we use Lemma 2.1 to conclude that
d ∂f
∂f
− ( 0 ) = 0.
∂y dx ∂y
(2.7)
This equation is called the Euler-Lagrange1 equation of second order. A function y(x) which
satisfies the Euler-Lagrange equation (2.7) and the imposed end-point conditions is by definition
the extremizing function for the integral (2.1).
Next, as a generalization of the previous theorem, we will state and prove a few more theorems.
1
Named in honor of Leonhard Euler (1707-1783) and Joseph Louis de Lagrange (1736-1813).
2. COMPUTATION OF CONSERVED DENSITIES
6
Theorem 2.2: The system of differential equations that must be satisfied by continuously differentiable functions y1 (x), y2 (x), . . . , yn (x) which extremize the integral
I=
x2
Z
x1
f (x, y1 , . . . , yn , y10 , . . . , yn0 ) dx,
(2.8)
and which achieve prescribed values at the end points of integration x1 and x2 (with x1 < x2 )
is given by
∂f
d ∂f
− ( 0 ) = 0, i = 1, 2, . . . , n.
∂yi dx ∂yi
Proof: We denote the extremizing functions by y1 (x), . . . , yn (x), and form the one-parameter
family of “comparison” functions
Yi (x) = yi (x) + ηi (x),
i = 1, 2, . . . , n,
(2.9)
where ηi are arbitrary differentiable functions for which
ηi (x1 ) = ηi (x2 ) = 0,
i = 1, 2, . . . , n,
(2.10)
and is the parameter of the family. By the same argument as in Theorem 2.1, we see that
the set of extremizing functions y1 (x), . . . , yn (x) are members of the family for all choices of
η1 , . . . , ηn when = 0. Thus, if we form the integral
I() =
Z
x2
x1
f (x, Y1 , . . . , Yn , Y10 , . . . , Yn0 ) dx
(2.11)
by replacing yi , yi0 in (2.8) by Yi , Yi0 , respectively, we have that I(0) is the extremum we sought.
Through (2.9), we have that
Yi0 (x) = yi0 (x) + ηi0 ,
i = 1, 2, . . . , n,
(2.12)
and if we differentiate (2.11), and use (2.9) and (2.12) we obtain
Z
0
I () =
x2
x1
!
∂f
∂f
∂f 0
∂f 0
η ) + ··· + (
η )
(
η1 +
ηn +
0 1
∂Y1
∂Y1
∂Yn
∂Yn0 n
dx.
(2.13)
dx = 0.
(2.14)
Now, from I0 (0) = 0, it follows that
0
I (0) =
Z
x2
x1
!
∂f
∂f
∂f
∂f
(
η1 + 0 η10 ) + · · · + (
ηn + 0 ηn0 )
∂y1
∂y1
∂yn
∂yn
Eq. (2.14) must hold for all choices of the functions ηi (i = 1, 2, . . . , n). In particular, it holds
for the special choice in which ηi (i = 2, . . . , n) are identically zero and η1 is still arbitrary,
consistent with (2.10). With this selection of ηi we integrate the second term of the remaining
integral (2.14) by parts, and use (2.10) to obtain
Z
x2
x1
!
∂f
d ∂f
− ( 0 ) η1 dx = 0.
∂y1 dx ∂y1
(2.15)
2. COMPUTATION OF CONSERVED DENSITIES
7
Since (2.15) holds for all η1 , we conclude from Lemma 2.1 that
∂f
d ∂f
− ( 0 ) = 0.
∂y1 dx ∂y1
(2.16)
With an analogous argument we derive the Euler-Lagrange equations similar to (2.16) but with
y1 replaced by y2 , y3 , etc. Combining these equations we have
∂f
d ∂f
− ( 0 ) = 0,
∂yi dx ∂yi
i = 1, 2, . . . , n.
(2.17)
This system of simultaneous Euler-Lagrange equations must be satisfied by the functions
y1 (x), . . . , yn (x) in order to make the integral (2.8) extremal.
Theorem 2.3: The differential equation that must be satisfied by the continuously differentiable
function y(x), extremizing the integral
I=
Z
x2
f (x, y, y 0 , . . . , y (n) ) dx,
(2.18)
x1
and where y, y 0 , . . . , y (n−1) satisfy prescribed values at x1 and x2 , is given by
d ∂f
d2 ∂f
dn ∂f
∂f
− ( 0 ) + 2 ( 00 ) + · · · + (−1)n n ( (n) ) = 0.
∂y dx ∂y
dx ∂y
dx ∂y
Proof: Similar to the previous cases, we form the “comparison” family
Y (x) = y(x) + η(x),
(2.19)
where η is an arbitrary differentiable function satisfying
η(xk ) = η 0 (xk ) = · · · = η (n−1) (xk ) = 0,
k = 1, 2,
(2.20)
and is the parameter of the family. Notice that, on differentiating (2.19), we obtain
Y (j) (x) = y (j) (x) + η (j) (x),
j = 1, 2, . . . , n,
(2.21)
where Y (j) (x) denotes the j th derivative of Y. Again, we apply the same argument: for = 0 the
extremizing function y(x) is a member of the family no matter the choice of η. If we differentiate
the integral
Z x2
I() =
f (x, Y, Y 0 , . . . , Y (n) ) dx,
(2.22)
x1
and use (2.19) and (2.21), we obtain
Z
0
I () =
x2
∂f
∂f 0
∂f (n)
η+
η + ··· +
η ) dx.
0
∂Y
∂Y
∂Y (n)
(2.23)
∂f
∂f
∂f
η + 0 η 0 + · · · + (n) η (n) ) dx = 0.
∂y
∂y
∂y
(2.24)
(
x1
Thus,
I 0 (0) =
Z
x2
x1
(
2. COMPUTATION OF CONSERVED DENSITIES
8
Using (2.20) successively one gets
Z
x2
x1
∂f (j)
∂f (j−1) x2 Z x2 d ∂f
x −
(
η dx =
η
)η (j−1) dx
1
∂y (j)
∂y (j)
x1 dx ∂y (j)
Z x2
d ∂f
( (j) )η (j−1) dx
= 0−
x1 dx ∂y
Z x2
d ∂f
(
= (−1)1
)η (j−1) dx
x1 dx ∂y (j)
!
Z x2 2
x
d
d ∂f
∂f
1
(j−2) 2
(j−2)
(
x −
= (−1)
)η
(
)η
dx
1
dx ∂y (j)
x1 dx2 ∂y (j)
!
Z x2 2
d
∂f
= (−1)1 0 −
(
)η (j−2) dx
x1 dx2 ∂y (j)
Z x2 2
d
∂f
= (−1)2
( (j) )η (j−2) dx
2
x1 dx ∂y
..
.
Z x2 j
∂f
d
( (j) )η dx.
= (−1)(j)
j
x1 dx ∂y
(2.25)
Now, substituting (2.25) into (2.24) except the first term we obtain
Z
x2
x1
n
d ∂f
d2 ∂f
∂f
∂f
n d
− ( 0 ) + 2 ( 00 ) + · · · + (−1)
( (n) ) η dx = 0.
n
∂y dx ∂y
dx ∂y
dx ∂y
!
(2.26)
Since the integral (2.26) must vanish for all η, with Lemma 2.1, we obtain the generalized
Euler-Lagrange equations of the form:
d ∂f
d2 ∂f
dn ∂f
∂f
− ( 0 ) + 2 ( 00 ) + · · · + (−1)n n ( (n) ) = 0.
∂y dx ∂y
dx ∂y
dx ∂y
(2.27)
Theorem 2.4: The system of differential equations that must be satisfied by the extremizing
functions of the integral
I=
Z
x2
x1
is
(n)
f (x, y1 , . . . , y1 , . . . , yn , . . . , yn(n) ) dx,
∂f
d ∂f
d2 ∂f
dn ∂f
− ( 0 ) + 2 ( 00 ) + · · · + (−1)n n ( (n) ) = 0,
∂yi dx ∂yi
dx ∂yi
dx ∂yi
i = 1, 2, . . . , n,
(2.28)
where yi (x) (i = 1, 2, . . . , n) are continuously differentiable functions satisfying prescribed values at the boundaries of integration.
Proof: The proof can easily be derived by combining the strategies of the proofs of Theorem
2.2 and Theorem 2.3.
Using vector notation, system (2.28) can be written in compact form:
L~y (f ) = ~0,
2. COMPUTATION OF CONSERVED DENSITIES
9
where ~y = [y1 , . . . , yn ]T , L~y (f ) = [Ly1 (f ), . . . , Lyn (f )]T , and ~0 = [0, . . . , 0]T .
In the literature [33],
n
d ∂
∂
d2 ∂
∂
n d
Lyi =
− ( 0 ) + 2 ( 00 ) + · · · + (−1)
( (n) ),
n
∂yi dx ∂yi
dx ∂yi
dx ∂yi
is called the Euler operator.
The following theorem is crucial because it is related to how we will use the Euler-Lagrange
equations in our algorithm.
(n)
Theorem 2.5: If f = f (x, y1 , . . . , y1 , . . . , yn , . . . , yn(n) ), then the corresponding Euler-Lagrange
d
g, where
equations are satisfied identically if and only if f =
dx
(n−1)
, . . . , yn , . . . , yn(n−1) ).
g = g(x, y1 , . . . , y1
Proof: To give an idea how this theorem is proved, we will consider the simplest case explicitly.
Then, by pointing out the ideas behind the derivation of the Euler-Lagrange equations, we will
complete the proof.
Suppose that f (x, y, y 0 ) is the derivative of g(x, y). Then the integral (2.1) is equal to the
difference of prescribed values at the end points, that is g(x2 , y2 ) − g(x1 , y1 ). So we have:
d
∂g ∂g 0
g=
+
y,
dx
∂x ∂y
∂f
∂2g
∂2g
=
+ 2 y0,
∂y
∂x∂y ∂y
∂g
∂f
=
,
0
∂y
∂y
!
!
∂f
d ∂g
∂2g
∂2g 0
=
=
+
y.
∂y 0
dx ∂y
∂x∂y ∂y 2
f =
d
dx
If we substitute these terms into the corresponding Euler-Lagrange equation, we see that (2.7)
is satisfied identically. Conversely, if (2.7) is satisfied identically, we have:
∂f
∂2f
∂ 2 f 0 ∂ 2 f 00
−
−
y − 0 2 y ≡ 0.
∂y ∂x∂y 0 ∂y∂y 0
∂y
Since the first three terms on the left contain at most the first derivative of y, the identity
∂2f
requires the coefficient
of y 00 to vanish identically. This is equivalent to stating that f
∂y 0 2
must be linear in y 0 ; that is, f = M (x, y) + N (x, y)y 0 . From (2.7), with this particular f , we
∂M
∂N
have
≡
. This is precisely the condition that f is the derivative of some function g(x, y).
∂y
∂x
As we mentioned before, for the generalization we use the idea behind the derivation
of EulerZ x2
x2
Lagrange equations. Suppose that f is the derivative of g. Then we have
f dx = g x .
x1
1
Thus, the result of the integration is independent of choices of yi ’s, since the yi ’s satisfy the
prescribed values at the end points. Hence, the Euler-Lagrange equations must be satisfied
identically. Conversely, if the Euler-Lagrange equations are satisfied identically, then the result
2. COMPUTATION OF CONSERVED DENSITIES
10
of integration has to be the same for all yi ’s, which means that the result is independent of the
choice of the path given by the yi (x)’s. It follows that the integrand is a derivative of some
function g. Similar results for higher dimensions are given in [33, pp. 252].
To show the usefulness of this simple Theorem in verifying if a given function is the derivative
of another function, let us consider an example.
Suppose that
2
2
3
f (y, y 0 , y 000 , y (4) ) = y 0 (x)y 00 (x) + 2y(x)y 00 (x)y (3) (x) − 3y 0 (x) y 00 (x)y (3) (x) − y 0 (x) y (4) (x)
is given. We chose f so that it is the derivative with respect to x of the function
2
3
g(y, y 0 , y 00 , y (3) ) = y(x)y 00 (x) − y 0 (x) y (3) (x).
d
g. Now, in order to see that Theorem 2.4 is indeed
dx
true, we explicitly compute all the relevant terms in (2.27). For this example n = 4;
therefore, we need:
Indeed, one can easily verify that f =
∂f
= 2y 00 (x)y (3) (x),
∂y
d ∂f
2
2
( 0 ) = 2y 00 (x)y (3) (x) − 6y 00 (x) y (3) (x) − 6y 0 (x)y (3) (x)
dx ∂y
2
−12y 0 (x)y 00 (x)y (4) (x) − 3y 0 (x) y (5) (x),
d2 ∂f
2
2
( 00 ) = 8y 00 (x)y (3) (x) − 6y 00 (x) y (3) (x) − 6y 0 (x)y (3) (x) + 6y 0 (x)y (4) (x)
2
dx ∂y
2
−12y 0 (x)y 00 (x)y (4) (x) + 2y(x)y (5) (x) − 3y 0 (x) y (5) (x),
d3 ∂f
2
2
( 000 ) = 8y 00 (x)y (3) (x) − 36y 00 (x) y (3) (x) − 18y 0 (x)y (3) (x) + 6y 0 (x)y (4) (x)
3
dx ∂y
2
−24y 0 (x)y 00 (x)y (4) (x) + 2y(x)y (5) (x) − 3y 0 (x) y (5) (x),
d4 ∂f
2
2
2
( (4) ) = −36y 00 (x) y (3) (x)−18y 0 (x)y (3) (x) −24y 0 (x)y 00 (x)y (4) (x)−3y 0 (x) y (5) (x).
4
dx ∂y
Substitution of the right hand sides into
∂f
d ∂f
d2 ∂f
d3 ∂f
d4 ∂f
− ( 0 ) + 2 ( 00 ) − 3 ( (3) ) + 4 ( (4) )
∂y dx ∂y
dx ∂y
dx ∂y
dx ∂y
(2.29)
upon simplification indeed gives zero.
2.2 Conservation Laws
For the partial differential equation
∆(x, t, u(x, t)) = 0,
(2.30)
2. COMPUTATION OF CONSERVED DENSITIES
11
where t ∈ IR, x ∈ IR are time and spatial variables, respectively, and u(x, t) ∈ IR is the
dependent variable, a conservation law is an equation of the form
Dt ρ + Dx J = 0,
(2.31)
which is satisfied for all solutions of (2.30), where ρ(x, t), the conserved density, and J(x, t),
associated flux, are, in general, functions of x, t, u and the partial derivatives of u; Dt denotes the
total derivative with respect to t; and Dx the total derivative with respect to x [1]. Furthermore,
ρ is a local conserved density if ρ is a local functional of u, i.e. if the value of ρ at any x depends
only on the values of u in an arbitrary small neighborhood of x. If J is also local, then (2.31) is
a local conservation law. In particular, if ρ is a polynomial in u and its x derivatives, and does
not depend explicitly on x or t, then ρ is called a polynomial conserved density. If J is also
such a polynomial, then (2.31) is called a polynomial conservation law. Thus, a polynomial
conservation law for a nonlinear PDE can be expressed in the form
ρt + Jx = 0.
(2.32)
There is a close relationship between constants of motion and conservation laws. Integration
of (2.32) yields a constant of local conservation type
P =
Z
+∞
ρ dx = constant,
−∞
provided that J vanishes at infinity.
The most famous evolution equation from soliton theory, the Korteweg-de Vries (KdV)
equation [29], ut + uux + u3x = 0, is known to have infinitely many polynomial conservation
laws. The first three polynomial conservation laws are given by:
1 2
+
u + u2x = 0,
2
x
2 3
+
u − ux 2 + 2uu2x = 0,
3
x
1 4
2
2
2
+ − u + 2uux − u u2x − u2x + 2ux u3x = 0.
4
x
(u)t
u2
1
− u3 + ux 2
3
t
t
The first two, express conservation of momentum and energy, respectively. They are easy to
compute by hand. The third one, which is less obvious, requires more work.
2.3 Algorithm
We now describe our algorithm to compute polynomial type conservation laws for systems of
nonlinear evolution equations.
In contrast to rather complicated algorithms [4, 7], we introduce a simpler algorithm which is
based in part on ideas presented in [21, 31] and [10, 41, 44]. Our algorithm consists of three
major steps:
• Determining the weights (scaling properties) of variables and parameters,
2. COMPUTATION OF CONSERVED DENSITIES
12
• Constructing the form of the density,
• Determining the unknown coefficients.
Consider a system of N evolution equations:
(1)
(n)
ui,t + Fi (uj , uj , . . . , uj ) = 0,
def
where ui,t =
i = 1, 2, . . . , N,
j = 1, 2, . . . , N,
∂ui (n) def ∂ n (ui )
, ui =
, and ui depend on x and t (i = 1, 2, . . . , N ).
∂t
∂xn
Remark 2.1: From our algorithm, it will be clear that we need an evolution equation for each
function ui that occurs in the system. So, in case we do not have an evolution equation for
each function (because there are more dependent variables than equations), we can always add
trivial evolution equations, ui,t = 0. For instance, if u2 is a function in the system, but we have
no evolution equation for u2 , we add u2,t = 0 to the system.
2.3.1 Determining the Weights of Variables and Parameters
We define the weight2 of a variable as the number of partial derivatives with respect to x that
variable carries, and the rank of a term as the total weight of that term in terms of partial
derivatives with respect to x [21, 31]. For a given system of evolution equations, we first try
to determine the weights (scaling properties) of the variables in the system. We assume that
all terms in an equation have the same rank, and we call this uniformity in rank. Different
equations in the same system can have different ranks.
Introducing some useful notations will make the explanation of the procedure for determining
the weights of variables much easier.
• w returns the weight of its argument.
For example, w(u1 ) is the unknown weight of variable u1 .
• g returns the degree of nonlinearity of its argument in a term.
For example, g(u1 ) = 3 if u1 occurs cubically in a term.
• d returns the number of partial derivatives with respect to its argument in a term. For
(2)
example, in u1 (1) (u2 )2 we have d(x) = 5.
• ri,k denotes the rank of the k th term in the ith equation.
For example, r1,3 is the rank of the third term in the first equation.
Let us look in more detail at another example. If we work with the term u1,t u2 2 u2 (2) , we have
g(u1 ) = 1, g(u2 ) = 3, d(x) = 2, and d(t) = 1.
Recall that we have defined the weight of a variable in terms of x−derivatives. Therefore, we
∂n
∂
(can) choose w( ) = 1, . . . , w( n ) = n.
∂x
∂x
In order to lead up towards the generalization, we start with the following example.
2
Weights are assumed to be nonnegative and rational.
2. COMPUTATION OF CONSERVED DENSITIES
13
Consider the KdV equation [29], to be discussed further in Chapter 3,
∂u
∂u ∂ 3 u
+u
+
= 0.
∂t
∂x ∂x3
(2.33)
Note that the KdV equation is invariant under the scaling symmetry
(x, t, u) → (λx, λ3 t, λ−2 u).
Hence, u carries the weight of two derivatives with respect to x, and t carries the weight of
three derivatives with respect to x. Fortunately, scaling properties can be determined in a
more algorithmic way. In our notation, (2.33) becomes
(3)
0
u1,t + u1 u1 0 + u000
1 = u1,t + u1 u1 + u1 = 0.
(2.34)
∂
∂
To determine the weight w(u1 ) of u1 and the weight w( ) of , form the expressions for
∂t
∂t
the ranks of the various terms. For this example we have:
r1,1 = 1 w(
∂
) + w(u1 )
∂t
for the term u1,t ,
r1,2 = 1 + 2 w(u1 )
for the term u1 u1 0 , and
r1,3 = 3 + 1 w(u1 )
(3)
for the term u1 . Then uniformity in rank requires that r1,1 = r1,2 = r1,3 . Solving this
∂
system for the unknowns w(u1 ) and w( ) gives
∂t
w(u1 ) = 2, w(
Alternatively, we could denote this by u1 ∼
∂
) = 3.
∂t
∂2
∂
∂3
,
and
∼
.
∂x2
∂t
∂x3
Next, we explain the procedure for general cases. Most of the time, only the variables ui and
∂
will have weights. However, to be able to handle some general systems with or without
∂t
constant parameters, we allow for constant parameters to be introduced and also for them to
carry weights. Let us assume that there are P such parameters in the system. We denote
these parameters by pi , i = 1, 2, ..., P. Thus, the extended list of variables that carry weights is
∂
{ ∂t
, u1 , u2 , . . . , uN , p1 , p2 , . . . , pP }. Then, we proceed with the following steps:
Step (1) Take the ith equation. Suppose it has Ki terms.
Step (2) For the k th term in the ith equation compute the rank based on the following expression:
ri,k = d(x) + d(t) w(
N
P
X
X
∂
)+
g(uj ) w(uj ) +
g(pj ) w(pj ),
∂t
j=1
j=1
k = 1, 2, . . . , Ki .
2. COMPUTATION OF CONSERVED DENSITIES
14
Recall that g returns the degree of nonlinearity of its argument in a term. If the variable uj
and/or the parameter pj is missing in a term, then g(uj ) = 0 or g(pj ) = 0, or both.
Step (3) Using uniformity in rank in the ith equation, form the linear system:
Ai = {ri,1 = ri,2 = · · · = ri,Ki }.
Step (4) Repeat steps (1) through (3) for all of the equations in the given system.
Step (5) Gather the equations Ai to form the global linear system: A =
N
[
Ai ,
i=1
∂
Step (6) Solve A for the unknowns w(uj )’s, w(pj )’s and w( ∂t
).
Let us consider the Modified Vector Derivative Nonlinear Schrödinger (MVDNLS) equation
[44], which can be written as a coupled system:
ut + u(u2 + v 2 ) + βu − vx
vt + v(u2 + v 2 ) + ux
x
x
= 0,
= 0.
(2.35)
In our notation, system (2.35) becomes
u1,t + βu1 0 + 3u1 2 u1 0 + u2 2 u1 0 + 2u1 u2 u2 0 − u2 00 = 0,
u2,t + 2u1 u2 u1 0 + u1 2 u2 0 + 3u2 2 u2 0 + u1 00 = 0.
(2.36)
The physics behind this system, as explained in Chapter 3 [44], requires that we give a
∂
weight w(β) to β, as well as to u1 , u2 and . Doing so, we have:
∂t
∂
) + 1 w(u1 ),
∂t
1 + 1 w(u1 ) + 1 w(β),
1 + 3 w(u1 ),
1 + 1 w(u1 ) + 2 w(u2 ),
1 + 1 w(u1 ) + 2 w(u2 ),
2 + 1 w(u2 ),
∂
1 w( ) + 1 w(u2 ),
∂t
1 + 2 w(u1 ) + 1 w(u2 ),
1 + 2 w(u1 ) + 1 w(u2 ),
1 + 3 w(u2 ),
2 + 1 w(u1 ).
r1,1 = 1 w(
r1,2
r1,3
r1,4
r1,5
r1,6
=
=
=
=
=
r2,1 =
r2,2
r2,3
r2,4
r2,5
=
=
=
=
Then, we form
A1 = {r1,1 = r1,2 = r1,3 = r1,4 = r1,5 = r1,6 },
A2 = {r2,1 = r2,2 = r2,3 = r2,4 = r2,5 },
A = A1 ∪ A2 .
2. COMPUTATION OF CONSERVED DENSITIES
15
∂
) and w(β). This gives
Next, we solve A for w(u1 ), w(u2 ), w( ∂t
∂
w(u1 ) = w(u2 ) = 1/2, w( ∂t ) = 2 and w(β) = 1. We denote this by
u1 2 ∼ u 2 2 ∼ β ∼
∂
∂
∂2
and
∼
.
∂x
∂t
∂x2
Thus far, we have assumed uniformity in rank for each equation. In the next example, we show
we can get around this restriction by introducing auxiliary parameters with (unknown) weights
in the equations.
Consider the Boussinesq equation [1],
utt − u2x + u u2x + u2x + a u4x = 0,
(2.37)
where a is a nonzero parameter. Although it is a wave equation, we can rewrite it as a
system of evolution equations:
u1,t + u2 0 = 0,
(3)
u2,t + u1 0 − u1 u1 0 − a u1
= 0.
(2.38)
It is easy to see that the second equation in the system (2.38) does not have uniform rank.
(3)
Indeed, the terms u1 0 and a u1 do not allow uniform rank. In order to be able to use our
algorithm, we introduce an auxiliary parameter b with a weight. So, the new system will
contain parameters a and b, with only b having weight. From the viewpoint of weights, we
treat these parameters as extra variables, but they are true parameters, therefore all their
derivatives with respect to either t or x are zero. Hence, we replace (2.38) by a new system:
u1,t + u2 0 = 0,
(3)
u2,t + b u1 0 − u1 u1 0 − a u1
= 0.
If we use the same procedure as before, we obtain:
∂
) + 1 w(u1 ),
∂t
1 + 1 w(u2 ),
∂
1 w( ) + 1 w(u2 ),
∂t
1 + 1 w(u1 ) + 1 w(b),
1 + 2w(u1 ),
3 + 1 w(u1 ).
r1,1 = 1 w(
r1,2 =
r2,1 =
r2,2 =
r2,3 =
r2,4 =
Then we form the systems
A1 = {r1,1 = r1,2 },
A2 = {r2,1 = r2,2 = r2,3 = r2,4 },
and A = A1 ∪ A2 .
∂
) and w(b) yields
Solving A for w(u1 ), w(u2 ), w( ∂t
w(u1 ) = 2, w(b) = 2, w(u2 ) = 3 and w(
∂
) = 2.
∂t
(2.39)
2. COMPUTATION OF CONSERVED DENSITIES
16
2.3.2 Constructing the Form of the Density
The second important step in our algorithm is the determination of the most general form of
the density with a prescribed rank. In other words, all the terms that appear in the density
need to have the same (given) rank. Notice that the rank of the density can be different from
any of the ranks of the equations in the system.
Suppose that we work with the KdV equation (2.34) and we try to construct the form of
the density with, say, rank 8. Since we have determined earlier that w(u1 ) = 2 for the KdV
equation, we can only have the following powers of u1 , producing rank 8 or less:
G = {(1; 0), (u1 ; 2), (u1 2 ; 4), (u1 3 ; 6), (u1 4 ; 8)},
where for each pair the second component denotes the total weight of the first one. Any
of the terms in the set G simply involve powers of u1 . We have not introduced derivatives
∂`
with respect to x yet. To create the terms that will have the desired rank, we apply
∂x`
to the first components of all of the pairs, where ` = 8, 6, 4, 2, 0, respectively. Notice that,
for each term, the number of derivatives, `, is such that it produces terms of exactly weight
8. For example, for the term u1 2 , which is already of weight 4, we need 4 extra derivatives.
Partial differentiation takes care of the distribution of the derivatives. Computing the terms
resulting from these differentiations, we obtain
∂ 8 (1)
→ {0},
∂x8
∂ 6 (u1 )
→ {u1 (6) },
∂x6
∂ 4 (u1 2 )
→ {(u1 (2) )2 , u01 u1 (3) , u1 u1 (4) },
4
∂x
Gathering these small sets results in
∂ 0 (u1 4 )
→ {u1 4 },
∂x0
∂ 2 (u1 3 )
→ {u1 (u1 0 )2 , u1 2 u1 (2) }.
2
∂x
H = {0, u1 (6) , (u1 (2) )2 , u01 u1 (3) , u1 u1 (4) , u1 (u1 0 )2 , u1 2 u1 (2) , u1 4 }.
Then we remove the terms from the set H that can be written as a derivative with respect
to x, or can be written as a derivative up to terms that were kept (prior) in the set. Note
that
u1 2 u1 (2) =
u01 u1 (3) =
u1 u1 (4) =
d
(u1 2 u01 ) − 2u1 (u1 0 )2 ,
dx
d
(u01 u1 (2) ) − (u1 (2) )2 ,
dx
d
(u1 u1 (3) − u01 u1 (2) ) +
dx
0 =
(6)
u1
=
(u1 (2) )2 .
d
(constant),
dx
d
(u1 (5) ),
dx
Indeed, removing the redundant terms from the set H, we end up with the final set
I = {(u1 (2) )2 , u1 (u1 0 )2 , u1 4 }.
With this simple procedure, we were able to find all the polynomial terms in u1 and its
x−derivatives such that (i) they have a fixed rank (8 here), and (ii) they cannot be written
as derivatives in x, or (iii) as derivatives modulo other terms in the set.
The density is then a linear combination with constant coefficients of the terms in the set
2
I. Hence, ρ = c1 (u1 (2) )2 + c2 u1 (u1 0 ) + c3 u1 4 .
2. COMPUTATION OF CONSERVED DENSITIES
17
Let us now turn to the general procedure. Suppose V = {v1 , v2 , . . . , vQ } is the sorted list of
∂
) that have positive weight. The variables are
all variables, including parameters (except
∂t
ordered such that w(v1 ) is the largest and w(vQ ) is the smallest weight. In order to determine
the form of the density of rank R, we use the following procedure:
Step (1) We form all the combinations of the variables in V that lead to rank R or less. Recursively, we form sets consisting of ordered pairs (Tq,s ; Wq,s ), where Tq,s denotes a combination of
different powers of the variables, and Wq,s denotes the total weight of Tq,s such that Wq,s ≤ R.
Set B0 = {(1; 0)} and proceed with the following.
For q = 1 through Q do
For m = 0 through M − 1 do
bq,m
Form Bq,m =
[
{(Tq,s ; Wq,s )}, where M is the number of pairs in Bq−1 , Tq,s =
s=0
Tq−1,m vq s , Wq,s = Wq−1,m + s w(vq ), (Tq−1,m ; Wq−1,m ) is the (m + 1)st ordered pair
q−1,m
]] is the maximum allowed power of vq .
in Bq−1 and bq,m = [[ R−W
w(vq )
Set Bq =
M[
−1
Bq,m .
m=0
Step (2) Set G = BQ . Notice that G has the all possible combinations of powers of the variables
that produce rank R or less.
Step (3) Now we introduce partial derivatives with respect to x. For each pair (TQ,s ; WQ,s ) of
∂`
G, apply
to the term TQ,s , if ` = (R − WQ,s ) is an integer, i.e., we introduce just enough
∂x`
partial derivatives so that all the pairs have weight R. Gathering the terms resulting from
∂ ` (TQ,s )
, we form the set H.
computing the various
∂x`
Step (4) Removing the terms from H that can be written as a derivative with respect to x,
or as a derivative up to terms kept (prior) in the set, we form the final set I, which consists of
the ‘building blocks’ of the density with the desired rank R.
Step (5) If I has I elements, which are the building blocks of ρ, then the linear combination
of these elements will produce the most general form of the density with rank R. Therefore,
ρ=
I
X
ci I(i),
i=1
where I(i) is the ith element in I, and ci ’s are the coefficients to be determined.
We return to the Boussinesq equation, represented by the system (2.39), where w(u1 ) =
2, w(b) = 2, and w(u2 ) = 3. Let us construct the density with rank R = 6. So, we have
V = {u2 , u1 , b}; hence v1 = u2 , v2 = u1 and v3 = b and, obviously, Q = 3. Let us go through
our procedure step by step:
Step (1)
2. COMPUTATION OF CONSERVED DENSITIES
18
For q = 1, m = 0:
b1,0 = [[ 63 ]] = 2. Thus, with T1,s = u2 s , W1,s = 3s and s = 0, 1, and 2 we obtain
B1 = B1,0 = {(1; 0), (u2 ; 3), (u2 2 ; 6)}.
For q = 2:
• With (1; 0), we have b2,0 = [[ 6−0
]] = 3. So, with
2
T2,s = 1 u1 s , W2,s = 0 + 2s = 2s, and s = 0, 1, 2, and 3,
we get
B2,0 = {(1; 0), (u1 ; 2), (u1 2 ; 4), (u1 3 ; 6)}.
• With (u2 ; 3), we obtain
B2,1 = {(u2 ; 3), (u1 u2 ; 5)},
since b2,1 = [[ 6−3
]] = 1, and T2,s = u2 u1 s , W2,s = 3 + 2s, s = 0, 1.
2
]] = 0. Therefore, B2,2 = {(u2 2 ; 6)}.
• The pair (u2 2 ; 6) yields to b2,2 = [[ 6−6
2
Hence,
B2 = {(1; 0), (u1 ; 2), (u1 2 ; 4), (u1 3 ; 6), (u2 ; 3), (u1 u2 ; 5), (u2 2 ; 6)}.
For q = 3:
Now, we introduce possible powers of b. An analogous procedure leads to
B3,0
B3,1
B3,2
B3,3
B3,4
B3,5
B3,6
=
=
=
=
=
=
=
{(1; 0), (b; 2), (b2 ; 4), (b3 ; 6)},
{(u1 ; 2), (bu1 ; 4), (b2 u1 ; 6)},
{(u1 2 ; 4), (bu1 2 ; 6)},
{(u1 3 ; 6)},
{(u2 ; 3), (bu2 ; 5)},
{(u1 u2 ; 5)},
{(u2 2 ; 6)}.
Thus,
B3 = {(1; 0), (b; 2), (b2 ; 4), (b3 ; 6), (u1 ; 2), (bu1 ; 4), (b2 u1 ; 6), (u1 2 ; 4),
(bu1 2 ; 6), (u1 3 ; 6), (u2 ; 3), (bu2 ; 5), (u1 u2 ; 5), (u2 2 ; 6)}.
Step (2) Set G = B3 .
Step (3) Now we attach derivatives to the first components of pairs in G. If we compute `,
defined in step (3), for each pair of G, we obtain
` = 6, 4, 2, 0, 4, 2, 0, 2, 0, 0, 3, 1, 1, 0,
2. COMPUTATION OF CONSERVED DENSITIES
19
respectively. Notice that in this case all ` values are integers. Therefore, we obtain
∂ 6 (1)
→ {0},
∂x6
∂ 4 (b)
→ {0},
∂x4
b3 → {b3 },
b2 u1 → {b2 u1 },
u1
3
3
→ {u1 },
∂ 4 (u1 )
→ {u1 (4) },
∂x4
∂ 2 (b2 )
→ {0},
∂x2
∂ 2 (bu1 )
→ {bu1 (2) },
∂x2
u2 2 → {u2 2 },
∂ 3 (u2 )
→ {u2 (3) },
3
∂x
bu1 2 → {bu1 2 },
∂(bu2 )
→ {bu2 0 },
∂x
∂(u1 u2 )
∂ 2 (u1 2 )
→ {u1 u2 0 , u1 0 u2 },
→ {(u1 0 )2 , u1 u1 (2) },
∂x
∂x2
where the terms in the list on the right hand sides come from explicitly computing the
partial derivatives. Gathering the sets on the right results in
2
H = {0, b3 , u1 (4) , bu1 (2) , b2 u1 , (u1 0 ) , u1 u1 (2) , bu1 2 , u1 3 , u2 (3) , bu2 0 , u1 u2 0 , u1 0 u2 , u2 2 }.
Step (4) Removing from H the terms that can be written as a derivative with respect to
x, or as a derivative up to terms retained earlier in the set, gives
2
I = {b2 u1 , bu1 2 , u1 3 , u2 2 , u1 0 u2 , (u1 0 ) },
since
0 =
(4)
u1
=
u1 u1 (2) =
bu2 0 =
d
(constant),
dx
d
(u1 (3) ),
dx
d
(u1 u1 0 ) − (u01 )2 ,
dx
d
(bu2 ),
dx
b3 =
bu1 (2) =
u2 (3) =
u1 u2 0 =
d
(b3 x),
dx
d
(bu1 0 ),
dx
d
(u2 (2) ),
dx
d
(u1 u2 ) −
dx
u1 0 u2 .
Step (5)
Therefore, the form of the density with rank 6 is
2
ρ = c1 b2 u1 + c2 bu1 2 + c3 u1 3 + c4 u2 2 + c5 u1 0 u2 + c6 (u1 0 ) .
2.3.3 Determining the Unknown Coefficients
Recall that a conservation law is of the form ρt + Jx = 0, or ρt = −Jx , which implies that ∂ρ
∂t
must be the partial derivative of some functional (−J) with respect to x. After computation
of ∂ρ
, we remove all (ui,t )(j) ’s from the expression using the evolution equations in the system.
∂t
This is done by solving the equations for ui,t first, and then computing their partial derivatives
with respect to x. The resulting expression for ∂ρ
must be the partial derivative of (−J) with
∂t
respect to x. There are two ways to compute J. We can either integrate the expression by
parts as many times as possible. If there remains a non-integrable part, it must vanish; this
will impose conditions on the coefficients ci . The second alternative is based on Theorem 2.5
from Section 2.1. This theorem, which requires that the Euler-Lagrange equations must vanish
identically, allows us to check whether or not an expression is a derivative. Again, if there is a
non-integrable part, we will make it vanish by the appropriate choice of the coefficients ci . No
matter which approach we use, we will have to solve a linear system for the coefficients ci .
2. COMPUTATION OF CONSERVED DENSITIES
20
For the KdV equation (2.34), the form of the density with rank 6 is
2
ρ = c1 u1 3 + c2 (u1 0 ) .
∂ρ
After replacement of u1,t and (u1,t )0 in
by −u1 u1 0 −u1 (3) and −(u1 u1 0 +u1 (3) )0 , respectively,
∂t
and integration with respect to x, we obtain
∂ρ
= 3c1 u1 2 u1,t + 2c2 u1 0 u1,t 0
∂t
= −3c1 u1 2 (u1 u1 0 + u1 (3) ) − 2c2 u1 0 (u1 u1 0 + u1 (3) )x
=
h
2
−(3/4)c1 u1 4 + (3c1 − c2 )u1 (u1 0 ) − 3c1 u1 2 u1 (2)
2
+c2 (u1 (2) ) − 2c2 u1 0 u1 (3)
i
3
x
− (3c1 + c2 )(u1 0 ) .
(2.40)
Obviously, the last term in (2.40) is non-integrable. We set it equal to zero and solve
3c1 + c2 = 0 for c1 . In c1 = − 31 c2 , we have the freedom of choosing c2 , since any density can
only be determined up to an arbitrary multiplicative coefficient. We choose c2 = 1. This
leads to
1
2
ρ = − u1 3 + (u1 0 ) ,
3
1
2
2
J = − u1 4 + 2u1 (u1 0 ) − u1 2 u1 (2) − (u1 (2) ) + 2u1 0 u1 (3) .
4
We are ready to generalize this procedure.
Step (1) After computing ∂ρ
, replace all (ui,t )(j) , i = 0, 1, .., N and j = 1, 2, ... from the
∂t
equations of the given system.
Step (2) The resulting expression, say E, must be the partial derivative of some functional
(−J) with respect to x. At this stage we have two options:
1. Carry out all integrations by parts, isolate the non-integrable part, and set it equal to
zero. The latter leads to a linear system for the coefficients ci .
2. Use the Euler-Lagrange equations in Theorems (2.4) and (2.5), and first apply the Euler
operator [33] L to E. If E is completely integrable no terms will be left, i.e. L~u (E) ≡ ~0,
where ~u = [u1 , . . . , un ]T with T representing the transpose; otherwise set the remaining
terms equal to zero and form the linear system for the coefficients ci .
With either option we construct a linear system, which we will denote by S.
Step (3) Two cases may occur, depending on whether or not there are parameters in the
system:
Case I: If the only unknowns in S are ci ’s, we just solve S for the ci ’s. Substitution of the
nonempty solution into the form of density gives the final form of the density.
Case II: If in addition to the coefficients ci ’s there are also parameters in S, then we determine
the conditions on the parameters so that a density in the given form exists for at least some
ci ’s nonzero. These compatibility conditions assure that the system has other than trivial
solutions. Solutions with all ci ’s zero lead to the trivial (zero) density which we ignore. Solving
2. COMPUTATION OF CONSERVED DENSITIES
21
the compatibility conditions may lead to different densities of the same rank, corresponding
to different choices of the parameters. Thus, generating the compatibility conditions enables
us to filter out all the cases for which there exists a nontrivial density of the given form. Let
{c1 , c2 , . . . , cI } be the set of all the coefficients that appear in the form of the density. In order
to determine all possible compatibility conditions, we proceed with the following procedure:
1. Set C = {c1 , c2 , . . . , cI }, and set n = 1.
2. While C =
6 {} do:
For the building block with coefficient cn to appear in the density, we must
have cn 6= 0. We can set cn = 1 and eliminate all the other ci ’s from S. This
gives compatibility conditions consistent with the presence of the term with
coefficient cn in the density.
If compatibility conditions require some of the parameters to be zero, then:
cn must be zero, since parameters are assumed to be nonzero. Set C = C\{cn },
and n = n0 , where n0 is the smallest index of the cj ’s that remain in C,
else:
(i) Solve the compatibility conditions and for each resulting branch:
• Solve the system S for ci ’s,
• Substitute the solution into the expression of the density to produce the final
form of the density.
(ii) Put the ci ’s which turn out to be zero for all of the branches into the set
Z. Since the ci ’s in Z might have not occurred in any of the densities yet, in
order to give them a chance to occur, we set C = C ∩ Z, and set n = n0 , where
n0 is the smallest index of the cj ’s that are still in C.
Consider the coupled KdV equations due to Hirota and Satsuma [12]:
1
ut − 3uux + 6vvx − u3x = 0,
2
vt + 3uvx + v3x = 0.
(2.41)
In our notation, system (2.41) can be rewritten as
1
u1,t − 3u1 u1 0 + 6u2 u2 0 − u1 (3) = 0,
2
0
u2,t + 3u1 u2 + u2 (3) = 0.
(2.42)
In the usual way, we determine the weights of u1 and u2 ; w(u1 ) = w(u2 ) = 2, and also
compute the form of the density of rank 6:
2
2
ρ = c1 u1 3 + c2 u1 2 u2 + c3 u1 u2 2 + c4 u2 3 + c5 (u1 0 ) + c6 (u2 0 ) + c7 u2 u1 (2) .
(2.43)
To determine the coefficients ci we first compute ∂ρ
, and then replace (u1,t ), (u2,t ), (u1,t )0 ,
∂t
(u2,t )0 , (u1,t )(2) from the equations in (2.42). This leads to
3
E = 9c1 u1 3 u1 0 + 6c2 u1 2 u2 u1 0 + 3c3 u1 u2 2 u1 0 + 6c5 (u1 0 )
2. COMPUTATION OF CONSERVED DENSITIES
22
−3c2 u1 3 u2 0 − (18c1 + 6c3 )u1 2 u2 u2 0 − (12c2 + 9c4 )u1 u2 2 u2 0
2
−6c3 u2 3 u2 0 − (12c5 + 6c6 )u1 0 (u2 0 ) + 6c5 u1 u1 0 u1 (2) + 9c7 u2 u1 0 u1 (2)
−3c7 u1 u2 0 u1 (2) − 12c5 u2 u1 0 u2 (2) − 6c6 u1 u2 0 u2 (2) − 18c7 u2 u2 0 u2 (2)
3c1 2 (3)
c3
+
u1 u1 + (c2 + 3c7 )u1 u2 u1 (3) + u2 2 u1 (3) − c2 u1 2 u2 (3)
2
2
(3)
2 (3)
−2c3 u1 u2 u2 − (3c4 + 6c7 )u2 u2 − c7 u1 (2) u2 (3) + c5 u1 0 u1 (4)
c7
−2c6 u2 0 u2 (4) + u2 u1 (5) ,
2
which must be a derivative. Using the fact that L~u (E) ≡ ~0 yields the following system of
equations S:
{c2 = 0,
2 c1 + c3 = 0, 4 c2 + 3 c4 = 0,
c3 − 4 c5 = 0,
c1 + 2 c5 = 0,
c3 − 12 c5 − 2 c6 = 0,
c3 + c6 = 0,
c7 = 0,
−c4 + c7 = 0,
c2 + 2 c7 = 0}.
Solving this linear system leads to the following solution set:
{c1 =
c3
−c3
, c4 = 0, c6 = −c3 , c2 = 0, c5 = , c7 = 0},
2
4
and by setting c3 = 1, we obtain
1
1
{c1 = − , c2 = 0, c3 = 1, c4 = 0, c5 = , c6 = −1, c7 = 0}.
2
4
After substitution of these values of the coefficients into the form of the density (2.43), we
obtain:
1
1
2
2
ρ = − u1 3 + u1 u2 2 + (u1 0 ) − (u2 0 ) .
2
4
Consider the family of fifth-order KdV equations,
ut + αu2 ux + βux u2x + γuu3x + u5x = 0,
(2.44)
where α, β, γ are non-zero parameters. In our notation, (2.44) becomes
u1,t + αu1 2 u1 0 + βu1 0 u1 (2) + γu1 u1 (3) + u1 (5) = 0.
(2.45)
Since w(u1 ) = 2, the form of the conserved density of rank 6 is
2
ρ = c1 u1 3 + c2 (u1 0 ) .
Replacing (u1,t ) and (u1,t )0 in
∂ρ
∂t
from the equation (2.45), we obtain:
3
2
E = −3αc1 u1 4 u1 0 − 4αc2 u1 (u1 0 ) − (3βc1 + 2αc2 )u1 2 u1 0 u1 (2) − 2βc2 u1 0 (u1 (2) )
2
−3γc1 u1 3 u1 (3) − 2(β + γ)c2 (u1 0 ) u1 (3) − 2γc2 u1 u1 0 u1 (4) − 3c1 u1 2 u1 (5) − 2c2 u1 0 u1 (6) .
2. COMPUTATION OF CONSERVED DENSITIES
23
Now E must be the derivative of some functional (−J). If we carry out all the integrations
by parts, we end up with the following non-integrable expression:
2
3
[−3βc1 + 9γc1 + 2αc2 ]u1 (u1 0 ) − [15c1 + 2βc2 − γc2 ]u1 0 (u1 (2) ) ,
which must vanish identically. Hence, the system of equations to be solved is
S = {−3βc1 + 9γc1 + 2αc2 = 0, −15c1 − 2βc2 + γc2 = 0}.
Since there are also parameters involved in the system, we have to compute the compatibility
conditions. We have C = {c1 , c2 }. If we let c1 = 1, we obtain
10α + 2β 2 − 7βγ + 3γ 2 = 0
as the compatibility condition. Solving this condition leads to
α=
−2β 2 + 7βγ − 3γ 2
,
10
and using it together with the choice c1 = 1, gives
{c1 = 1, c2 =
15
}.
γ − 2β
Substituting this solution into the given form of the density results in
ρ = u1 3 +
15
2
(u1 0 ) .
γ − 2β
None of the ci ’s are found to be zero. Thus, Z = {} and C = {}. Therefore, the procedure
of finding compatibility conditions ends here. The result that we obtained in this example
involves equations due to Lax, Sawada and Kotera, Caudry, Dodd and Gibbon, and Kaup
and Kuperschmidt. Several well-known equations belong to the the fifth-order KdV family
(2.44). We will return to this important topic in Chapter 3.
Still working with (2.44), we will determine the compatibility conditions and the corresponding densities of rank 8. In this case, the density has the form
2
2
ρ = c1 u1 4 + c2 u1 (u1 0 ) + c3 (u1 00 ) .
Going through the same steps as in the previous examples, we obtain the following system:
S = {3βc1 − 12γc1 − αc2 = 0, −30c1 − βc2 + 2γc2 + 5αc3 = 0, 5c2 + 2βc3 + γc3 = 0}.
Setting c1 = 1 leads to
45α(β − 2γ) = −(β − 4γ)(2β + γ)(β − 2γ),
as a compatibility condition. Solving this compatibility condition yields
2. COMPUTATION OF CONSERVED DENSITIES
24
6γ
6
2
2
u1 (u1 0 ) + (u1 00 ) .
α
γ
2
2
−2β + 7βγ + 4γ
135
675
2
2
• α=
and ρ = u1 4 −
u1 (u1 0 ) +
(u1 00 ) .
45
2β + γ
(2β + γ)2
• β = 2γ and ρ = u1 4 −
So, in this example, we obtain different densities of the same rank due to the choices for
the parameters. The first branch corresponds to equations due to Lax and Ito, while the
second corresponds to equations due to Sawada and Kotera, and Kaup and Kuperschmidt,
and also Ito. All these equations belong to the fifth-order KdV family (see Chapter 3).
Consider the parametrized coupled KdV equations due to Hirota and Satsuma [12]:
ut − 6αuux + 6vvx − αu3x = 0,
vt + 3uvx + v3x = 0.
(2.46)
In our notation, system (2.46) can be rewritten as
u1,t − 6αu1 u1 0 + 6u2 u2 0 − αu1 (3) = 0,
u2,t + 3u1 u2 0 + u2 (3) = 0.
(2.47)
We know that w(u1 ) = w(u2 ) = 2 and the form of the density of rank 4 is
ρ = c 1 u 1 2 + c 2 u 1 u2 + c 3 u2 2 .
Going through steps (1) and (2) of the procedure we obtain:
S = {(1 + α)c2 = 0, 2 c1 + c3 = 0}, and C = {c1 , c2 , c3 }.
• c1 = 1 leads to the solution
{c1 = 1, c2 = 0, c3 = −2}
without any constraints on the parameter α. Since only c2 turned out to be zero, we set
Z = {c2 } and therefore C = {c2 } with n0 = 2.
• c2 = 1 leads to the compatibility condition α = −1 and the solution
1
{c1 = − c3 , c2 = 1}.
2
Since Z = {}, the procedure ends here. Therefore, we obtained two densities of the same
rank, one without any constraint. We can summarize the situation as follows:
• ρ = u1 2 − 2 u2 2 , and
• ρ = − 21 c3 u1 2 + u1 u2 + c3 u2 2 with the compatibility condition α = −1.
We end this rather long chapter with yet another example.
2. COMPUTATION OF CONSERVED DENSITIES
25
This time we consider a fabricated system:
ut − αuux − βu3x = 0,
vt − γvux − δuvx + vvx = 0.
In our notation, the above system can be rewritten as
u1,t − αu1 u1 0 − βu1 (3) = 0,
u2,t − γu2 u1 0 − δu1 u2 0 + u2 u2 0 = 0.
It is easy to determine that w(u1 ) = w(u2 ) = 2. The form of the density of rank 4 is
ρ = c 1 u 1 2 + c 2 u 1 u2 + c 3 u2 2
as in the previous example. Going through steps (1) and (2) of the procedure gives the
following system of equations to be solved:
S = {βc2 = 0, (α + γ − 2δ) c2 = 0, c2 + 4γc3 − 2δc3 = 0}, and C = {c1 , c2 , c3 }.
• c1 = 1 leads to the solution
{c1 = 1, c2 = 0, c3 = 0}
without any constraints on the parameters. Since c2 and c3 turned out to be zero, we set
Z = {c2 , c3 } and therefore C = {c2 , c3 } with n0 = 2.
• c2 = 1 leads to the compatibility condition β = 0. Since parameters are assumed to be
nonzero, we cannot have c2 = 1; therefore c2 has to be zero. We set
C = C\{c2 } = {c3 } and n0 = 3.
• c3 = 1 leads to the compatibility condition
1
γ = δ,
2
with {c2 = 0, c3 = 1} as a solution. Since at this point C becomes the empty set, we end
the procedure. In conclusion, we have
• ρ = u1 2 without any constraint on the parameters, and
• ρ = c1 u1 2 + u2 2 provided that γ = 12 δ.
The algorithm described in this chapter is implemented in Mathematica. Details about the
program CONDENS.M are in Chapter 4. æ
3. APPLICATIONS AND EXAMPLES
For systems with parameters, our algorithm can be used as a tool to find the necessary conditions on the parameters to have a density of fixed rank. If for some choices of the parameters
the system admits (infinitely) many conservation laws, we may have detected an integrable
system. This is the major application of our program.
Another application relates to numerical analysis, where the knowledge of conserved densities
may be used to check the accuracy of numerical integration algorithms. The first section of
this chapter discusses these applications. In the second section, we will list the results for many
test cases.
3.1 Applications
3.1.1 Fifth-Order Korteweg-de Vries Equation
Consider the fifth-order Korteweg-de Vries equation:
ut + αu2 ux + βux u2x + γuu3x + u5x = 0,
(3.1)
where α, β, γ are nonzero parameters. Special cases of this family are given in the literature
[5, 6, 13, 14, 19, 24, 38, 39]:
Lax Case
Sawada − Kotera (SK) Case
(or Caudry − Dodd − Gibbon Case)
Kaup − Kupershmidt (KK) Case
Ito Case
α = 30, β = 20, γ = 10,
α = 5, β = 5, γ = 5,
α = 20, β = 25, γ = 10,
α = 2, β = 6, γ = 3.
∂
∂5
∂2
,
∼
.
The scaling properties of (3.1) indicate that u ∼
∂x2
∂t
∂x5
Using our algorithm, we easily computed the compatibility conditions for the parameters α, β
and γ so that this equation admits a polynomial type conserved density of fixed rank. We list
our results:
Rank 2: There are no conditions on the parameters. This is not surprising since equation
(3.1) is a conservation law. The corresponding density is
ρ = u.
Rank 4: We have the density
ρ = u2
3. APPLICATIONS AND EXAMPLES
27
provided that the condition
β = 2γ
(3.2)
is satisfied. Therefore, only the Lax and Ito cases admit a density of rank 4.
Rank 6: The condition
10α = −2β 2 + 7βγ − 3γ 2
(3.3)
must hold in order to have a density of rank 6, which is then
ρ = u3 +
15
ux 2 ,
(−2β + γ)
γ 6= 2β.
Therefore, the Lax, SK and KK equations admit a density of rank 6.
Rank 8: Here we have two branches:
1. If the condition
β = 2γ
(3.4)
is satisfied, we have
6
6γ
uux 2 + u2x 2 .
α
α
This branch covers the Lax and Ito cases.
ρ = u4 −
2. If the condition
α=
−2β 2 + 7βγ + 4γ 2
45
(3.5)
holds, then we have
ρ = u4 −
135
675
uux 2 +
u2x 2 ,
2β + γ
(2β + γ)2
γ 6= −2β.
This branch covers the SK, KK and Ito cases.
Rank 10: The conditions
β = 2γ and 10α = 3γ 2
(3.6)
must be satisfied. This leads to the density
ρ = u5 −
500
50 2 2 100
u ux + 2 uu2x 2 − 3 u3x 2 .
γ
γ
7γ
Therefore, only the Lax equation admits density of rank 10.
Now, the following question arises: What are the necessary conditions on the parameters α, β
and γ so that the fifth-order KdV equation (3.1) could admit infinitely many polynomial conservation laws?
Lax Case: We found that the Lax equation admits densities of ranks 4 and 6. Combining the
conditions (3.2) and (3.3) leads to the conditions
α=
3 2
γ and β = 2γ.
10
(3.7)
3. APPLICATIONS AND EXAMPLES
28
In (3.7), α and β are fixed in terms of γ. This refers to the Lax case which is known to be
integrable.
SK-KK Cases: We found that the SK and KK equations admit densities of ranks 6 and 8.
Combining the conditions (3.3) and (3.5) gives the constraints
1
α = γ 2 and β = γ,
5
1 2
5
α = γ and β = γ.
5
2
(3.8)
(3.9)
According to the literature, (3.8) corresponds to the SK equation, and (3.9) to the KK equation. Both equations are indeed integrable.
Ito Case: We found that the Ito equation admits only three densities. Combining the conditions (3.4) and (3.5) gives
2γ 2
α=
and β = 2γ.
(3.10)
9
Condition (3.10) corresponds to the Ito equation which is not integrable. Therefore, using our
algorithm, we were able to filter out the known integrable cases of the family of fifth-order KdV
equations.
3. APPLICATIONS AND EXAMPLES
29
Tab. 3.1: Conserved Densities for Sawada-Kotera and Lax 5th-Order Equations
Density
Sawada-Kotera equation
Lax equation
ρ1
u
u
ρ2
—
1 2
u
2
ρ3
1 3
u
3
− ux 2
1 3
u
3
− 16 ux 2
ρ4
1 4
u
4
− 94 uux 2 + 34 u2x 2
1 4
u
4
− 12 uux 2 +
ρ5
—
1 5
u
5
− u2 ux 2 + 15 uu2x 2 −
ρ6
1 6
u
6
1 6
u
6
− 53 u3 ux 2 −
−
25 3
u ux 2
4
+2u2x 3 −
ρ7
ρ8
1 7
u
7
−
17
u 4
8 x
21
uu3x 2
8
− 9u4 ux 2 −
+ 6u2 u2x 2
+ 38 u4x 2
54
uux 4
5
489
uu2x 3
35
u u 2+
− 288
35 2x 3x
81
uu4x 2
35
—
57 3
u u2x 2
5
+
+ 648
u 2u 2 +
35 x 2x
−
−
5
+ 63
u2x 3 −
261 2
u u3x 2
35
9
u 2
35 5x
1 7
u
7
5
u 4
36 x
1
uu3x 2
14
+
1
u 2
70 3x
+ 21 u2 u2x 2
1
u 2
252 4x
− 52 u4 ux 2 − 56 uux 4 + u3 u2x 2
+ 12 ux 2 u2x 2 +
10
uu2x 3
21
5
− 42
u2x u3x 2 +
1 8
u
8
1
u 2
20 2x
−
1
uu4x 2
42
− 72 u5 ux 2 −
3 2
u u3x 2
14
1
u 2
924 5x
−
35 2
u ux 4
12
+ 74 u4 u2x 2
+ 72 uux 2 u2x 2 + 53 u2 u2x 3 +
7
u 4
24 2x
+ 12 u3 u3x 2 − 14 ux 2 u3x 2 − 56 uu2x u3x 2
1 2
+ 12
u u4x 2 +
1
+ 3432
u6x 2
7
u u 2
132 2x 4x
−
1
uu5x 2
132
3. APPLICATIONS AND EXAMPLES
30
3.1.2 Seventh-Order Korteweg-de Vries Equation
Consider the family of seventh-order Korteweg-de Vries equations:
ut + au3 ux + bux 3 + cuux u2x + du2 u3x + eu2x u3x + f ux u4x + guu5x + u7x = 0,
(3.11)
where a, b, c, d, e, f, g are nonzero parameters. Special cases of this family are [14]:
SK − Ito Case a = 252, b = 63, c = 378, d = 126, e = 63, f = 42, g = 21,
Lax Case
a = 140, b = 70, c = 280, d = 70, e = 70, f = 42, g = 14.
The scaling properties of (3.11) are easily determined, and from them we know that
∂2
u∼
,
∂x2
∂7
∂
∼
.
∂t
∂x7
Again, with our algorithm we compute the compatibility conditions for the parameters, so that
certain conserved densities will exist. Here are the results:
Rank 2: The family has density
ρ=u
provided that the condition
c − 2d
2
is satisfied. Both the Lax and SK-Ito equations satisfy this condition.
Rank 4: The conditions
b = c − 3d and e = 5(f − 2g)
b=
(3.12)
(3.13)
have to be satisfied. If so, the family has density
ρ = u2 .
Only the Lax equation admits this density.
Rank 6: We have three branches. If one of the following three sets of conditions is satisfied:
1
(28bf − 42cf + 120f 3 − 42bg + 63cg − 720f 2 g + 1350f g 2 − 810g 3 ),
294
14c
5
d =
(2f 2 − 9f g + 9g 2 ) and e =
,
(3.14)
14
2f − 3g
or
1
a =
(4bf − 6cf + 24df − 6bg + 9cg − 36dg),
42
14c − 14d + 10f 2 − 45f g + 45g 2
e =
,
(3.15)
2f − 3g
or
1
(28bf − 42cf − 40f 3 − 42bg + 63cg + 240f 2 g − 450f g 2 + 270g 3 ),
a =
294
5
2(21c + 20f 2 − 90f g + 90g 2 )
2
2
d =
(−2f + 9f g − 9g ) and e =
,
(3.16)
42
3(2f − 3g)
a =
3. APPLICATIONS AND EXAMPLES
31
Tab. 3.2: Conserved Densities for Kaup-Kupershmidt and Ito 5th-Order Equations
Density
Kaup-Kupershmidt equation
Ito equation
ρ1
u
u
ρ2
—
u2
2
ρ3
u3
3
− 18 ux 2
ρ4
u4
4
−
ρ5
—
ρ6
u6
6
9
uux 2
16
u7
7
+
−
35 3
u ux 2
16
−
31
u 4
256 x
−
15
uu3x 2
128
27 4
u ux 2
8
− 171
u u 2+
640 2x 3x
—
+
+
51 2
u u2x 2
64
—
3
u 2
512 4x
369
uux 4
320
−
+ 2619
u 2u 2 +
4480 x 2x
ρ8
u4
4
3
u 2
64 2x
—
37
+ 256
u2x 3 −
ρ7
—
2211
uu2x 3
2240
27
uu4x 2
560
69 3
u u2x 2
40
+
−
−
—
477 2
u u3x 2
1120
9
u 2
4480 5x
—
− 49 uux 2 + 34 u2x 2
3. APPLICATIONS AND EXAMPLES
32
Tab. 3.3: Conserved Densities for Sawada-Kotera-Ito and Lax 7th-Order Equations
Density
Sawada-Kotera-Ito equation
Lax equation
ρ1
u
u
ρ2
—
u2
ρ3
−u3 + ux 2
−2u3 + ux 2
ρ4
3u4 − 9uux 2 + u2x 2
5u4 − 10uux 2 + u2x 2
ρ5
—
−14u5 + 70u2 ux 2 − 14uu2x 2 + u3x 2
ρ6
− 12
u6 +
7
150 3
u ux 2
7
+
− 16
u 3 + uu3x 2 −
21 2x
ρ7
17
u 4
7 x
163
uu2x 3
9
u u 2 + uu4x 2 −
− 32
9 2x 3x
—
48 2
u u2x 2
7
1
u 2
21 4x
−
− 73 u6 +
70 3
u ux 2
3
+
− 10
u 3 + uu3x 2 −
9 2x
5u7 − 105u4 ux 2 − 42uux 4 +
+24ux 2 u2x 2 +
ρ8
−
133 3
u u2x 2
3
29 2
u u3x 2
3
1
u 2
27 5x
− 23 u7 +
35 4
u ux 2
3
− 73 ux 2 u2x 2 −
5
u u 2
9 2x 3x
3 8
u
2
+
35
u 4
18 x
− 7u2 u2x 2
1
u 2
18 4x
35
uux 4
9
20
uu2x 3
9
− 19 uu4x 2 +
−
14 3
u u2x 2
3
+ u2 u3x 2
1
u 2
198 5x
− 42u5 ux 2 − 35u2 ux 4 + 21u4 u2x 2
+42uux 2 u2x 2 + 20u2 u2x 3 + 72 u2x 4
−6u3 u3x 2 − 3ux 2 u3x 2 − 10uu2x u3x 2
+u2 u4x 2 +
1
+ 286
u6x 2
7
u u 2
11 2x 4x
−
1
uu5x 2
11
3. APPLICATIONS AND EXAMPLES
then we have
ρ = u3 −
21
ux 2 ,
2f − 3g
33
3
f 6= g.
2
Rank 8: The density is
252(2f − g)
1764
uux 2 −
u2x 2 ,
2
2
−49c − 10f + 45f g − 20g
−49c − 10f 2 + 45f g − 20g 2
49c 6= −10f 2 + 45f g − 20g 2 ,
ρ = u4 +
provided that the conditions
1
a =
(−49cf − 10f 3 + 196cg + 85f 2 g − 200f g 2 + 80g 3 ),
882
7c − 2f 2 + 9f g − 4g 2
14c + 2f 2 − 9f g + 4g 2
, d=
and
b =
42
42
e = 2f − g
(3.17)
are satisfied.
Now, combining the conditions through Rank 2 and Rank 8 gives
4 3
1
6
2
a=
g , b = g 2 , c = g 2 , d = g 2 , e = 3g, f = 2g,
(3.18)
147
7
7
7
or
5
9
2
7
4 3
g , b = g 2 , c = g 2 , d = g 2 , e = 6g, f = g,
(3.19)
a=
147
14
7
7
2
or
5
5
10
5
a = g 3 , b = g 2 , c = g 2 , d = g 2 , e = 5g, f = 3g,
(3.20)
98
14
7
14
Notice that, in each case all of the parameters are fixed in terms of g. Condition (3.18) leads
to the SK-Ito equation, and condition (3.20) corresponds to the Lax equation, both equations
are integrable.
Another Integrable Case?
Interestingly enough and to the best of our knowledge, there seems to be no mention in the
literature of the equation that corresponds to the case (3.19). To compute the densities for this
case, we simply set g = 42. This gives
a = 2016, b = 630, c = 2268, d = 504, e = 252, f = 147, g = 42.
We computed the first five densities:
ρ1 = u,
ρ2 = −8u3 + ux 2 ,
1
4
ρ3 = − u4 + uux 2 − u2x 2 ,
3
36
31
102 2 2 37 3
1
192 6
ρ4 = −
u + 168u3 ux 2 + ux 4 −
u u2x − u2x + uu3x 2 − u4x 2 ,
5
10
5
30
60
480 7 3780 4 2 861
644
291
737
ρ5 = −
u +
u ux +
uux 4 −
u3 u2x 2 −
ux 2 u2x 2 −
uu2x 3
53
53
106
53
212
318
133
2
1
+u2 u3x 2 +
u2x u3x 2 − uu4x 2 +
u5x 2 .
636
53
1908
3. APPLICATIONS AND EXAMPLES
34
and conjecture that the equation
ut + 2016u3 ux + 630ux 3 + 2268uux u2x + 504u2 u3x +
252u2x u3x + 147ux u4x + 42uu5x + u7x = 0,
(3.21)
is also integrable. Further study of other properties of the equation is needed to prove the
conjecture. This is beyond the scope of this thesis.
3.1.3 Hirota-Satsuma System
Hirota and Satsuma [12] proposed a coupled KdV equation which describes interactions of two
long waves with different dispersion relations. The coupled system is given by
ut − 6αuux + 6vvx − αu3x = 0,
vt + 3uvx + v3x = 0,
(3.22)
where α is a nonzero parameter. It is known that (3.22) is completely integrable for α = 21 .
The scaling properties of (3.22) are such that
∂
∂3
∂2
,
∼
.
∂x2
∂t
∂x3
Our search for the compatibility conditions gives the following results:
Rank 2: There is no condition on the parameter α. One always has the trivial density
u∼v∼
ρ = u.
Rank 4: At this level, two branches emerge:
1. There is no condition for the density
ρ = u2 − 2v 2 .
2. If α = −1 then we have
1
ρ = uv + c (v 2 − u2 ),
2
c free.
Rank 6: There is no condition that fixes α, and we have the following density:
3
1
3
ρ = u3 −
uv 2 − ux 2 +
vx 2 , α 6= −1.
α+1
2
α+1
Rank 8: The system admits the density
12
12
24
4
1
8
ρ = u4 − u2 v 2 + v 4 − 2uux 2 − uvx 2 − v 2 u2x + u2x 2 + v2x 2 ,
5
5
5
5
5
5
1
provided that α = 2 . Therefore, we see that as soon as we try to compute the density of rank
8, we have to fix α, and exactly that value of α leads to the one and only one integrable case
of the Hirota-Satsuma system.
With α = 21 , we computed another density:
ρ =
−7 5
7
1
u + u3 v 2 − uv 4 + u2 ux 2 + v 2 ux 2 + u2 vx 2 + 4v 2 vx 2 + uv 2 u2x + vx 2 u2x
20
4
2
2
1
7
1
− uu2x 2 − 2uv2x 2 + u3x 2 + v3x 2 + v 2 u4x .
20
40
5
10
3. APPLICATIONS AND EXAMPLES
35
3.1.4 Testing Numerical Algorithms for PDEs
When a partial differential equation that models wave-like phenomena is to be solved numerically, it is highly desirable that the discretization conserves the discrete analogues of the
conserved quantities for the equation. In particular, the conservation of a positive definite
quadratic quantity in some cases rules out the occurrence of nonlinear instabilities.
Most of the time, the construction of schemes conserving quadratic quantities is confined to
the semidiscrete case, i.e., discretization occurs only in space variables, while the time is kept
continuous so as to approximate the original partial differential equation by a system of ordinary
differential equations. However, the solution of this semidiscrete system must be obtained by
a numerical method for ordinary differential equations, and, as a result, integration in time
may cause the loss of conservation properties of the semidiscrete approximation. In fact, the
time integration results in failure of the attempt to conserve quadratic quantities for an explicit
method.
In [11], the use of conservation laws in solving the Boussinesq equation has been described. In
[37], Sanz-Serna describes a scheme for the integration in time of partial differential equations
which is explicit and capable of conserving quadratic functionals conserved by the semidiscrete
approximation. For the Korteweg-de Vries equation, we know that u and u2 are conserved
densities. We mentioned earlier that
P =
Z
+∞
ρ dx = constant,
−∞
provided J vanishes at infinity. So, a discrete scheme for the Korteweg-de Vries equation should
P
P
have the momentum j Ujn and the energy j [Ujn ]2 as conserved quantities. In [37], an explicit
self-adaptive conservative scheme with conservation of energy,
X
j
2
[Ujn−2 ] =
X
[Ujn ]2 ,
j
and momentum is explained. Conservation of energy implies boundedness of the solutions,
and therefore obviates the occurrence of blowup phenomena. More details about numerical
applications of conservation laws can be found in [25].
3.2 Examples
3.2.1 Korteweg-de Vries (KdV) Equation
In 1895, Korteweg and de Vries provided a simple analytic foundation for the study of solitary
waves by deriving the equation (now known by their names),
ut + uux + u3x = 0,
(3.23)
for shallow water waves which includes both nonlinear and dispersive effects but ignores dissipation [29, 32, 40]. The KdV equation has also arisen in a number of other physical contexts
such as stratified internal waves, ion-acoustic waves, plasma physics and lattice dynamics.
Conservation laws of the KdV equation are widely studied in [18, 21, 29, 31], and it has been
shown that there are infinitely many polynomial conservation laws. The scaling properties of
3. APPLICATIONS AND EXAMPLES
36
∂2
∂3
∂
∼
,
. We used our algorithm (and the computer program
∂x2
∂t
∂x3
CONDENS.M) to compute a set of polynomial conserved densities, of which the first eight
are:
(3.23) indicate that u ∼
ρ1 = u,
ρ2 = u 2 ,
1
ρ3 = − u 3 + u x 2 ,
3
1
3
ρ4 = − u4 + uux 2 − u2x 2 ,
12
5
1 5 5 2 2
3
ρ5 =
u − u ux + uu2x 2 − u3x 2 ,
36
6
7
7 6 35 3 2
35 4 7 2 2 10 3
1
ρ6 = −
u + u ux +
ux − u u2x − u2x + uu3x 2 − u4x 2 ,
648
54
108
6
9
3
7
7
10
3
1 7 35 4 2 35
u − u ux − uux 4 + u3 u2x 2 + ux 2 u2x 2 + uu2x 3 − u2 u3x 2
ρ7 =
216
72
36
6
2
3
2
3
−5u2x u3x 2 + uu4x 2 − u5x 2 ,
11
1 8
7 5 2 35 2 4
7
10
7
ρ8 =
u − u ux − u ux + u4 u2x 2 + 7uux 2 u2x 2 + u2 u2x 3 + u2x 4
864
36
36
12
3
2
42
6
18
u6x 2 .
−u3 u3x 2 − 3ux 2 u3x 2 − 10uu2x u3x 2 + u2 u4x 2 + u2x u4x 2 − uu5x 2 +
11
11
143
Remark 3.1: As of today, ρ = xu − 12 tu2 , is the only known non-polynomial conserved density
of the KdV equation.
3.2.2 Modified Korteweg-de Vries (MKdV) Equation
Replacing the quadratic nonlinearity in the KdV equation by a cubic one, leads to
ut + 6u2 ux + u3x = 0,
(3.24)
which is known as the modified Korteweg-de Vries equation [30]. The relevance of this equation
in applications is well-known. For instance, it has been used to describe acoustic waves in
anharmonic lattices and Alfvén waves in collisionless plasma. The scaling properties of (3.24)
∂
∂
∂3
reveal that u ∼
,
∼
. It is also known that MKdV equation has infinitely many
∂x
∂t
∂x3
polynomial conservation laws. Here we list six of them:
ρ1 = u,
ρ2 = u2 ,
ρ3 = −u4 + ux 2 ,
1
1
ρ4 = − u6 + u2 ux 2 − u2x 2 ,
5
10
5 8
1
1
ρ5 =
u − 5u4 ux 2 − ux 4 + u2 u2x 2 − u3x 2 ,
14
2
14
3. APPLICATIONS AND EXAMPLES
37
7
70
133 2 4
17
20
ρ6 = − u10 + u6 ux 2 +
u ux − 7u4 u2x 2 − ux 2 u2x 2 − uu2x 3
9
3
9
3
9
1
+u2 u3x 2 − u4x 2 .
18
3.2.3 Generalized Schamel (GS) Equation
The generalized Schamel equation is given by
n2 ut + (n + 1)(n + 2)u2/n ux + u3x = 0,
(3.25)
where n is a positive integer [41]. Obviously, for n = 4, (3.25) reduces to
16ut + 30u1/2 ux + u3x = 0,
(3.26)
which is a rescaled version of the Schamel equation that appears in the literature,
ut + u1/2 ux + αu3x = 0,
governing the propagation of ion-acoustic waves in a cold-ion plasma where some of the electrons
do not behave isothermally during the passage of the wave but are trapped in it. The square
root in the nonlinear term then translates to lowest order some of the kinetic effects, associated
with electron trapping, which dominate over the fluid effects giving rise themselves to the wellstudied KdV or MKdV equations. Rescaling the coefficients of (3.25) leads to the KdV equation
for n = 2, and the MKdV equation for n = 1. In [41] it has been reported that for n 6= 1, 2, the
equation (3.25) has only three polynomial conservation laws which are
ρ1 = u,
ρ2 = u2 ,
2
1
n2
ρ3 = ux 2 − u2+ n .
2
2
(3.27)
√
Here, we give the results for (3.26). We introduce the transformation u → u in order to
obtain an equation that is polynomial in u and its partial derivatives. Doing so, we obtain
16uut + 30u2 ux + 3ux u2x + uu3x = 0,
∂2
where u ∼
,
∂x2
densities:
(3.28)
∂
∂3
. We could find no more than three polynomial type conserved
∼
∂t
∂x3
ρ1 = u2 ,
ρ2 = u4 ,
ρ3 = −4u5 + u2 ux 2 .
Notice that these are the transformed forms of the densities in (3.27) with n = 4.
3. APPLICATIONS AND EXAMPLES
38
3.2.4 Ito System
Ito proposed the following coupled nonlinear wave equation [1, 15, 20]:
ut − 6uux − 2vvx − u3x = 0,
vt − 2vux − 2uvx = 0,
(3.29)
which differs from the Hirota-Satsuma system in the interaction and dispersion terms for v. In
the absence of v, system (3.29) reduces to the KdV equation. It is a Hamiltonian system with
infinitely many conservation laws. The scaling properties of the system (3.29) are u ∼ v ∼
∂3
∂2
∂
∼
,
, and the first five densities are:
∂x2
∂t
∂x3
ρ1 = c1 u + c2 v,
ρ2 = u2 + v 2 ,
ρ3 = −2u3 − 2uv 2 + ux 2 ,
3
1
1
1
1
ρ4 = − u4 − u2 v 2 − v 4 + uux 2 − v 2 u2x − u2x 2 ,
2
5
10
5
10
7 5
3 4 7 2 2 1 2 2 1 2 2
7
3 2
ρ5 =
u + u v + uv − u ux + v ux − v vx + uv 2 u2x + uu2x 2
10
10
2
2
5
10
1
1 2
2
− u3x + v u4x .
20
10
3.2.5 Nonlinear Schrödinger (NLS) Equation
The nonlinear Schrödinger (NLS) equation [1],
iqt − q2x + 2|q|2 q = 0,
(3.30)
plays a ubiquitous role in physics. It arises as an asymptotic limit of a slowly varying dispersive
wave envelope in a nonlinear medium and as such has significant applications in nonlinear optics,
water waves, and plasma physics. Equation (3.30) is known to be completely integrable, and
together with the KdV equation, is one of the most studied equations in soliton theory. Our
program cannot handle complex equations, but, by introducing q = u + iv, equation (3.30) can
be replaced by
ut − v2x + 2v(u2 + v 2 ) = 0,
vt + u2x − 2u(u2 + v 2 ) = 0.
Here the scaling properties of the system (3.31) are such that
u∼v∼
∂
,
∂x
∂
∂2
∼
.
∂t
∂x2
We list seven of the densities as computed with our program:
ρ1 = u 2 + v 2 ,
ρ2 = vux ,
(3.31)
3. APPLICATIONS AND EXAMPLES
39
ρ3 = u4 + 2u2 v 2 + v 4 + ux 2 + vx 2 ,
1
1
ρ4 = u2 vux + v 3 ux − vu3x ,
3
6
1 6 3 4 2 3 2 4 1 6 5 2 2 1 2 2 3 2 2 5 2 2
ρ5 = − u − u v − u v − v − u u x − v u x − u v x − v v x
2
2
2
2
2
2
2
2
1
1
+uv 2 u2x − u2x 2 − v2x 2 ,
4
4
1 2 3
3
1
1
1
3 4
ρ6 = − u vux − u v ux − v 5 ux + vux 3 − vux vx 2 + uvux u2x + u2 vu3x
4
2
20
4
4
4
1 3
1
+ v u3x − vu5x ,
12
40
5 8
15
5
35
5
ρ7 =
u + 5u6 v 2 + u4 v 4 + 5u2 v 6 + v 8 + u4 ux 2 − 5u2 v 2 ux 2 + v 4 ux 2
4
2
4
2
2
7 4 15 4 2
35
5
7
− ux + u vx + 25u2 v 2 vx 2 + v 4 vx 2 − ux 2 vx 2 − vx 4 − 10u3 v 2 u2x
4
2
2
2
4
1
5
7
7
−5uv 4 u2x − 5uvx 2 u2x + u2 u2x 2 + v 2 u2x 2 + u2 v2x 2 + v 2 v2x 2 − v 2 ux u3x
2
2
2
2
1 2 1 2
+ u3x + v3x + uv 2 u4x .
4
4
3.2.6 Derivative Nonlinear Schrödinger (DNLS) Equation
The derivative nonlinear Schrödinger (DNLS) equation was first derived by Rogister in 1971 for
the nonlinear evolution of parallel Alfvén waves in plasmas. Later it appeared in many other
different physical contexts such as quantum field theory, weakly nonlinear dispersive water
waves, and nonlinear optics [1, 44]. The well-studied DNLS is also completely integrable. Here
we consider a system-form of the DNLS:
ut + 3u2 ux + v 2 ux + 2uvvx − v2x = 0,
vt + 2uvux + u2 vx + 3v 2 vx + u2x = 0,
where u2 ∼ v 2 ∼
∂
,
∂x
(3.32)
∂
∂2
∼
. The first four conserved densities are:
∂t
∂x2
ρ1 = c1 u + c2 v,
ρ2 = u 2 + v 2 ,
1 4
1
ρ3 =
u + u2 v 2 + v 4 + 2vux ,
2
2
1 6 1 4 2 1 2 4
1
1
1
1
ρ4 =
u + u v + u v + v 6 + u2 vux + v 3 ux + ux 2 + vx 2 .
12
4
4
12
3
6
6
3.2.7 Modified Vector Derivative Nonlinear Schrödinger (MVDNLS)
Equation
The DNLS equation could also account for slightly oblique propagation of Alfvén waves, albeit
at the price of neglecting two effects which may be important in strongly magnetized astrophysical plasmas [44]. One is the deviation from charge neutrality between the different plasma
species, the other is the influence of the displacement current in Ampère’s law. Retaining these
3. APPLICATIONS AND EXAMPLES
40
effects results in a nonlinear vector evolution equation which differs from the standard vector
form of the DNLS by an extra linear term, and therefore is called the modified vector derivative
nonlinear Schrödinger (MVDNLS) equation. Obviously, the MVDNLS includes the DNLS as
a special case, when we go from oblique to parallel propagation, which amounts to dropping
the bothersome extra term. In [44], it has been shown that the MVDNLS possesses a biHamiltonian structure, and hence through the resulting recursion operator an infinite number
of conserved densities. The MVDNLS is, after the necessary scaling and Galilean transforms
to cast it in its simplest dimensionless form, given by
∂
∂B⊥
∂ 2 B⊥
∂B⊥
2
+
(B⊥
B⊥ ) + αB⊥0 B⊥0 ·
+ ex ×
= 0,
∂t
∂x
∂x
∂x2
(3.33)
where the parameter α characterizes the extra term which distinguishes the MVDNLS from
the DNLS. B⊥ stands for the perpendicular magnetic field, which includes both the wave
contributions and the static perpendicular field B⊥0 due to the oblique propagation with respect
to the total external magnetic field. The direction of wave propagation is along the x-axis. We
replace the vector equation by
ut + u(u2 + v 2 ) + βu − vx
vt + v(u2 + v 2 ) + ux
x
x
= 0,
= 0,
(3.34)
2
.
where u and v denote the components of B⊥ parallel and perpendicular to B⊥0 and β = αB⊥0
Note that B⊥0 is the norm of the vector B⊥0 . Checking the scaling properties of (3.34) gives
∂
∂
∂2
∂
, β ∼
,
∼
. Using the software, we computed the first six
that u2 ∼ v 2 ∼
∂x
∂x
∂t
∂x2
densities. They are:
ρ1 = c1 u + c2 v,
ρ2 = u2 + v 2 ,
1 2
ρ3 =
(u + v 2 )2 − uvx + ux v + βu2 ,
2
1 2
1
β
ρ4 =
(u + v 2 )3 + (ux 2 + vx 2 ) − u3 vx + v 3 ux + (u4 − v 4 ),
4
2
4
1 2
2
4
ρ5 =
(u + v 2 )4 − (ux v2x − u2x vx ) + (uux + vvx )2
4
5
5
6 2
+ (u + v 2 )(ux 2 + vx 2 ) − (u2 + v 2 )2 (uvx − ux v)
5
β
β2
+ (2ux 2 − 4u3 vx + 2u6 + 3u4 v 2 − v 6 ) + u4 ,
5
5
7 2
1
5
ρ6 =
(u + v 2 )5 + (u2x 2 + v2x 2 ) − (u2 + v 2 )(ux v2x − u2x vx )
16
2
2
15
+5(u2 + v 2 )(uux + vvx )2 + (u2 + v 2 )2 (ux 2 + vx 2 )
4
β
+ (5u8 + 10u6 v 2 − 10u2 v 6 − 5v 8 + 20u2 ux 2 − 12u5 vx + 60uv 4 vx − 20v 2 vx 2 )
8
35
β2
+ (u6 + v 6 ) − (u2 + v 2 )3 (uvx − ux v).
4
16
3. APPLICATIONS AND EXAMPLES
41
Based on the knowledge of the form of the densities, Willox et al. [44] were able to show that
the MVDNLS indeed admits a bi-Hamiltonian formulation. Without the explicit forms of the
densities this might have been a much harder task.
3.2.8 Kaup System
We consider a model for dispersive water waves given by [1, 3, 26]:
ut − 6uux + 3v 2 ux + 6uvvx − 3ux vx − 3vu2x + u3x = 0,
vt − 6vux − 6uvx + 3v 2 vx + 3vx 2 + 3vv2x + v3x = 0.
(3.35)
By the reduction v = 0, system (3.35) reduces to the KdV equation. Based on the scaling
∂2
∂
∂3
∂
properties of (3.35), we see that u ∼
,
∼
,
v
∼
. For the Kaup system, the
∂x2
∂x
∂t
∂x3
first six conserved densities are:
ρ1
ρ2
ρ3
ρ4
=
=
=
=
v,
u,
uv,
u2 − uv 2 + vux ,
3
ρ5 = −3u2 v + uv 3 − v 2 ux + vu2x ,
2
ρ6 = −2u3 + 6u2 v 2 − uv 4 − 6uvux + 2v 3 ux − ux 2 + uvx 2 − 2v 2 u2x + vu3x .
3.2.9 Dispersiveless Long Wave System
A dispersiveless long wave system is given as [1]
ut + vux + uvx = 0,
vt + ux + vvx = 0,
a system which is integrable. Here, scaling properties give that
u free,
We chose u ∼
v free, but u ∼ 2v.
∂
∂
and v 2 ∼
, to obtain the following set of densities:
∂x
∂x
ρ1 = v,
ρ2 = u,
ρ3 = uv,
ρ4 = u2 + uv 2 ,
ρ5 = 3u2 v + uv 3 ,
1 3
1
ρ6 =
u + u2 v 2 + uv 4 ,
3
6
1
ρ7 = u3 v + u2 v 3 + uv 5 ,
10
1 4
1
ρ8 =
u + 2u3 v 2 + u2 v 4 + uv 6 .
3
15
(3.36)
3. APPLICATIONS AND EXAMPLES
42
We also realized that the set of densities does not change with the choice of the scaling properties.
3.2.10 Broer-Kaup System
One of the most significant dispersive generalizations of the classical dispersiveless long wave
equations is the Broer-Kaup [23] system:
1
ut − uux − vx + u2x = 0,
2
1
vt − vux − uvx − v2x = 0,
2
(3.37)
where v and u are the height and the horizontal component of the velocity, respectively, of the
free surface. System (3.37) is integrable, that is, it has an infinite number of conservation laws.
The scaling properties of (3.37) indicate that
u∼
∂
,
∂x
v∼
∂2
,
∂x2
∂
∂2
∼
.
∂t
∂x2
Here is the list of the first seven conserved densities:
ρ1
ρ2
ρ3
ρ4
ρ5
ρ6
ρ7
=
=
=
=
u,
v,
uv,
−u2 v − v 2 + vux ,
1
1
= − u3 v − uv 2 + uvux − vu2x ,
3
3
1 4
3 2 2 1 3 3 2
3
3
1
1
=
u v + u v + v − u vux − v 2 ux + vux 2 − vx 2 + uvu2x − vu3x ,
4
2
2
2
4
4
4
4
3
3
1
1 5
u v + u3 v 2 + uv 3 − u3 vux − uv 2 ux + uvux 2 − uvx 2 + u2 vu2x
=
10
2
2
2
1
1
1 2
+ v u2x − vux u2x − uvu3x + vu4x .
2
2
10
3.2.11 Drinfel’d-Sokolov System
For the so-called Drinfel’d-Sokolov system [1]:
ut + 3vvx = 0,
vt + vux + 2uvx + 2v3x = 0,
∂2
∂
∂3
,
∼
, and the first five conserved densities:
∂x2
∂t
∂x3
= u,
= v2,
1
3
2
= − u3 + uv 2 + ux 2 − vx 2 ,
9
6
2
we have u ∼ v ∼
ρ1
ρ2
ρ3
(3.38)
3. APPLICATIONS AND EXAMPLES
43
1 4 9 2 2 27 4 9
27
45
3
81
u − u v − v − uux 2 + uvx 2 − v 2 u2x + u2x 2 − v2x 2 ,
2
4
16
4
2
8
8
8
1 4 1 2 2 1 2 2 7 2 2 1 2
5 2
=
uv + v ux − u vx − v vx + uv u2x − vx u2x
8
8
2
8
4
4
1
1
+uv2x 2 − v3x 2 + v 2 u4x .
2
4
ρ4 =
ρ5
3.2.12 Boussinesq Equation
The wave equation
utt − u2x + 3uu2x + 3ux 2 + au4x = 0,
(3.39)
was proposed by Boussinesq to describe surface water waves whose horizontal scale is much
larger than the depth of the water [1, 11]. Conservation laws play a key role in the study of the
Boussinesq equation because they restrict the set of possible solutions. In [11], conservation
laws are used to prove that solutions of the Boussinesq equation are bounded for a certain set
of initial conditions. Conservation laws are also used to prove that solutions fail to exist after
a finite time for another set of initial conditions.
For the purpose of deriving the conservation laws, we rewrite (3.39) as a system of first-order
equations by introducing a second dependent variable v:
ut + vx = 0,
vt + ux − 3uux − au3x = 0.
(3.40)
It is easy to see that the second equation in system (3.40) does not have uniform rank. Indeed,
ux and au3x
do not allow uniform rank. So we introduce an auxiliary parameter b with weight, and replace
the system (3.40) by
ut + vx = 0,
vt + bux − 3uux − au3x = 0.
(3.41)
Now we find that the scaling properties of (3.41) are such that
u∼b∼
∂2
,
∂x2
v∼
∂3
,
∂x3
∂
∂2
∼
.
∂t
∂x2
Nothing stops us from computing the conserved densities, of which the first four are:
ρ1
ρ2
ρ3
ρ4
=
=
=
=
u,
v,
uv,
bu2 − u3 + v 2 + aux 2 .
Now, if we substitute b = 1 into these densities we get the densities of (3.40) even though we
did not have uniformity in rank. This trick can be used in other circumstances where equations
in a system lack uniformity in rank.
3. APPLICATIONS AND EXAMPLES
44
3.2.13 3-Component Korteweg-de Vries Equation
Consider the 3-component extension of the KdV equation:
ut − 6uux + 2vvx + 2wwx − u3x = 0,
vt − 2vux − 2uvx = 0,
wt − 2wux − 2uwx = 0.
(3.42)
It is a bi-Hamiltonian system with an infinite number of conservation laws [22]. The scaling
∂2
∂3
∂
properties of (3.42) are: u ∼ v ∼ w ∼
∼
,
, and the first five densities are
∂x2
∂t
∂x3
ρ1 = c1 u + c2 v + c3 w,
ρ2 = u 2 − v 2 − w 2 ,
ρ3 = −2u3 + 2uv 2 + 2uw2 + ux 2 ,
5
1
1
ρ4 = − u4 + 3u2 v 2 − v 4 + 3u2 w2 − v 2 w2 − w4
2
2
2
1
+5uux 2 + v 2 u2x + w2 u2x − u2x 2 ,
2
3
3
3
7 5
ρ5 = − u + u3 v 2 − uv 4 + u3 w2 − uv 2 w2 − uw4
10
10
5
10
7 2 2 1 2 2 1 2 2 1 2 2 1 2 2
+ u ux + v ux + w u x + v v x − w v x
2
2
2
5
5
7
1 2 2
+ w wx + uv 2 u2x + uw2 u2x − uu2x 2
5
10
1
1 2
1
1
− vw v2x + u3x 2 + v 2 u4x + w2 u4x .
5
20
10
10
3.2.14 2-Component Nonlinear Schrödinger Equation
Consider the 2-component nonlinear Schrödinger equation [28, 48],
iqk,t − qk,2x + 2|q|2 qk = 0,
k = 1, 2.
(3.43)
By introducing q1 = u + iw, q2 = v + iz, equation (3.43) becomes
ut + 2(u2 w + v 2 w + w3 + wz 2 ) + w2x
vt + 2(u2 z + v 2 z + w2 z + z 3 ) + z2x
wt − 2(u3 + uv 2 + uw2 + uz 2 ) − u2x
zt − 2(u2 v + v 3 + vw2 + vz 2 ) − v2x
=
=
=
=
System (3.44) has infinitely many conserved densities [16], and
u∼v∼w∼z∼
∂
,
∂x
∂
∂2
∼
.
∂t
∂x2
0,
0,
0,
0.
(3.44)
3. APPLICATIONS AND EXAMPLES
We list the first four densities:
ρ1 = c1 (u2 + w2 ) + c2 (uz − vw) + c3 (uv + wz) + c4 (v 2 + z 2 ),
ρ2 = wux + zvx ,
1
1
1
ρ3 = u4 + u2 v 2 + v 4 + u2 w2 + v 2 w2 + w4 + u2 z 2
2
2
2
1
1
1
1
1
v 2 z 2 + w2 z 2 + z 4 − ux 2 − vx 2 − wx 2 − zx 2 ,
2
2
2
2
2
2
1
ρ4 = 2u2 wux + v 2 wux + w3 ux + uvzux + wz 2 ux + uvwvx + u2 zvx
3
2
2
1
1
1
+2v 2 zvx + w2 zvx + z 3 vx + vwzwx − uz 2 wx + wu3x + zv3x .
3
2
3
3
æ
45
4. USING THE PROGRAM CONDENS.M
In this chapter, we describe the features of our program CONDENS.M. The program is
written in Mathematica [46] and automatically carries out the tedious calculations needed to
determine conserved densities for systems of nonlinear evolution equations. The program CONDENS.M has its own menu interface which makes its use very easy and transparent. Apart
from having access to Mathematica, users need to have the program and the data files ready in
the same directory. We describe how to use the program by answering possible questions.
Remark 4.1: If you choose to use variational derivative instead of integration by parts, you also
need to have the file variation.m ready in the same directory. With our test cases, we noticed
that using variational derivative works faster.
Question 4.1: How do I start the program?
Once Mathematica comes up with ‘In[1]:=’ , type
In[1]:= <<condens.m
to read in the code condens.m. Then, via its menu interface, the program will automatically
prompt you for answers.
Question 4.2: How can I compute the density of rank 4 for the Drinfel’d-Sokolov system (3.38),
provided a conserved density of that rank exists?
Since the Drinfel’d-Sokolov system is stored in the menu, you just have to start the program,
answer some questions, and then choose the correct entry in the menu (22 in this case).
In[1]:= <<condens.m
.
.
.
*** MENU INTERFACE *** (page: 3)
------------------------------------------21) Kaup-Broer System (d_broer.m)
22) Drinfel’d-Sokolov System (d_soko.m)
23) Dispersiveless Long Wave System (d_disper.m)
24) 3-Component KdV System (d_3ckdv.m)
25) 2-Component Nonlinear Schrodinger Equation (d_2cnls.m)
26) Boussinesq System (d_bous.m)
nn) Next Page
tt) Your System
qq) Exit the Program
4. USING THE PROGRAM CONDENS.M
47
-----------------------------------------ENTER YOUR CHOICE: 22
Enter the rank of rho: 4
Use Variational Derivative instead of Integration by Parts? (y/n): y
Enter the name of the output file: d_soko4.o
*********************************************************
WELCOME TO THE MATHEMATICA PROGRAM
by UNAL GOKTAS and WILLY HEREMAN
FOR THE COMPUTATION OF CONSERVED DENSITIES OF
Drinfel’d-Sokolov System
Version 2.2 released on February 29, 1996
Copyright 1996
*********************************************************
.
.
*******************************************************
2
This is the density: u[2][x, t]
*******************************************************
2
(1,0)
2
This is the flux: 2 u[1][x, t] u[2][x, t] - 2 (u[2])
[x, t] +
(2,0)
>
4 u[2][x, t] (u[2])
[x, t]
*******************************************************
Result of explicit verification (rho_t + J_x) = 0
*******************************************************
In[2]:=
At the end of computation, you have access to the density and the flux. To see the density,
type
In[2]:= rho[x,t]
2
Out[2]= u[2][x, t]
To see the flux, type
In[3]:= j[x,t]
4. USING THE PROGRAM CONDENS.M
48
2
(1,0)
2
Out[3]= 2 u[1][x, t] u[2][x, t] - 2 (u[2])
[x, t] +
>
(2,0)
4 u[2][x, t] (u[2])
[x, t]
Question 4.3: How can I test my own systems?
By preparing a data file similar to the ones stored in the menu, one can test systems that are not
in the menu. Of course, the name for your data file should not coincide with the name of any
of the data files mentioned in the menu, unless you intentionally wanted to modify these data
files. For example, for the parametrized Hirota and Satsuma system (3.22), we have prepared
the data file d phrsat.m with the following lines:
(* start of data file d_phrsat.m *)
debug = False;
(* Hirota-Satsuma System *)
eq[1][x,t] = D[u[1][x,t],t]-aa*D[u[1][x,t],{x,3}]6*aa*u[1][x,t]*D[u[1][x,t],x]+6*u[2][x,t]*D[u[2][x,t],x];
eq[2][x,t] = D[u[2][x,t],t]+D[u[2][x,t],{x,3}]+
3*u[1][x,t]*D[u[2][x,t],x];
noeqs = 2;
name = "Hirota-Satsuma System (parametrized)";
parameters = {aa};
weightpars = {};
formrho[x,t] = {};
(* end of data file d_phrsat.m *)
Let us explain what the various lines in this data file mean:
debug = False;
Set it equal to True in order to see a detailed trace of the output.
eq[k][x,t] = ...;
Give the k th equation of the system in Mathematica notation.
Note that there is no == 0 at the end.
noeqs = 2;
This specifies the number of equations in the system.
name = "Hirota-Satsuma System (parametrized)";
Pick a short name for your system. Do not forget the quotes.
parameters = {aa};
4. USING THE PROGRAM CONDENS.M
49
Give a list of the parameters that are in the system. If there are no parameters, set parameters
= { };.
weightpars = {};
Give a list of the parameters that are supposed to have a weight (or dimension). Note that
weighted parameters are listed in weightpars, but not in parameters. The latter is a list of
dimensionless parameters only (see d bous.m and d mvdnls.m for examples).
formrho[x,t] = {};
If you want to give the form of rho, it has to be given here; otherwise, the program will compute
it. This allows you to test a form of rho obtained from the literature. It also allows you to
build up the form of rho, and compute undetermined coefficients.
Anything within (* and *) are comments, and they are ignored by Mathematica.
Once the data file is ready you can run it from the menu.
*** MENU INTERFACE *** (page: 1)
------------------------------------------1) KdV Equation (d_kdv.m)
2) Modified KdV Equation (d_mkdv.m)
3) Fifth Order KdV Equation-parametrized (d_5kdv.m)
4) Fifth Order KdV Equation-Lax Case (d_5lax.m)
5) Fifth Order KdV Equation-SK Case (d_5sk.m)
6) Fifth Order KdV Equation-KK Case (d_5kk.m)
7) Fifth Order KdV Equation-Ito Case (d_5ito.m)
8) Seventh Order KdV Equation-parametrized (d_7kdv.m)
9) Seventh Order KdV Equation-Lax Case (d_7lax.m)
10) Seventh Order KdV Equation-SK-Ito Case (d_7ski.m)
nn) Next Page
tt) Your System
qq) Exit the Program
-----------------------------------------ENTER YOUR CHOICE: tt
Make sure that you have prepared the data file for the system
that you want to test (similar to the supplied data files).
If your file is ready, press 1, else 2: 1
Enter the name of your data file: d_phrsat.m
Enter the rank of rho: 4
Use Variational Derivative instead of Integration by Parts? (y/n): y
Enter the name of the output file: d_phrst4.o
.
.
Question 4.4: Can I give the form of the density from outside?
Yes, that is possible. For example, include the line
formrho[x,t] = {c[1]*u[1][x,t]^3+c[2]*D[u[1][x,t],x]^2};
4. USING THE PROGRAM CONDENS.M
50
in the data file d kdv.m and run it from the menu. The braces are essential. The form of
the density must be given in the expanded form and with coefficients c[i]. In this case, the
program will skip the computation of the scaling properties of the system and the construction
of the form of ρ. Instead, it will just work with the given form of the density, and compute the
coefficients c[i]. In most cases, where you want to search for densities of specific rank, you must
set formrho[x,t] = { };.
Question 4.5: Do you allow parameters in the system?
The system can have one or more parameters. All parameters are assumed to be nonzero. For
examples, see d 5kdv.m, d 7kdv.m, d phrsat.m. If there are parameters in the system, the
program will compute the conditions for these parameters such that the density (of a given
rank) exists.
Question 4.6: Do you allow parameters which have a weight (dimension)?
The system can have one or more parameters with (unknown) weight, which makes it possible
to test systems with non-uniform rank. In case of doubt, you can start with parameters without
weight. If this causes an incompatibility in the assignment of ranks, the program may provide
a suggestion, and recommend that one or more parameters should be moved from the list
parameters into the list weightpars. For example, if you run d bous.m with the following lines
parameters = {beta,aa};
weightpars = {};
the program will proceed as follows:
*** MENU INTERFACE *** (page: 3)
------------------------------------------21) Kaup-Broer System (d_broer.m)
22) Drinfel’d-Sokolov System (d_soko.m)
23) Dispersiveless Long Wave System (d_disper.m)
24) 3-Component KdV System (d_3ckdv.m)
25) 2-Component Nonlinear Schrodinger Equation (d_2cnls.m)
26) Boussinesq System (d_bous.m)
nn) Next Page
tt) Your System
qq) Exit the Program
-----------------------------------------ENTER YOUR CHOICE: 26
Enter the rank of rho: 1
Use Variational Derivative instead of Integration by Parts? (y/n): y
Enter the name of the output file: d_bous1.o
.
.
.
In the given system there is at least one equation with terms
of unequal rank. Scaling properties can not be determined for
this system. We will try to figure out where the conflict is.
4. USING THE PROGRAM CONDENS.M
51
And, if possible, provide suggestions.
(1,0)
(3,0)
The terms beta (u[1])
[x, t] and -(aa (u[1])
[x, t])
in equation 2 are incompatible. Try to introduce an
auxiliary parameter with weight as coefficient of one of
these terms. Aborting the computation !
Out[6]= $Aborted
Following the suggestion, modify the data file by setting
parameters = {aa};
weightpars = {beta};
and, then repeat the search for densities.
Question 4.7: What happens if the given system is such that some of the weights are free?
In this case, the program will ask you to pick values for the free weights. For example, if you
run d disper.m you will encounter that situation:
.
.
.
ENTER YOUR CHOICE: 23
Enter the rank of rho: 1
Use Variational Derivative instead of Integration by Parts? (y/n): y
Enter the name of the output file: d_dspr1.o
*********************************************************
WELCOME TO THE MATHEMATICA PROGRAM
by UNAL GOKTAS and WILLY HEREMAN
FOR THE COMPUTATION OF CONSERVED DENSITIES OF
Dispersiveless Long Wave System
Version 2.2 released on February 29, 1996
Copyright 1996
*********************************************************
This is the equation 1:
(0,1)
(1,0)
(u[1])
[x, t] + u[2][x, t] (u[1])
[x, t] +
(1,0)
>
u[1][x, t] (u[2])
[x, t] = 0
This is the equation 2:
(0,1)
(1,0)
(1,0)
(u[2])
[x, t] + (u[1])
[x, t] + u[2][x, t] (u[2])
[x, t] = 0
Highest order of derivative in the system is 1.
Solve::svars: Warning: Equations may not give solutions for all "solve"
4. USING THE PROGRAM CONDENS.M
52
variables.
For the given system:
* weight on u[1] is 2 weightu[2].
* weight on u[2] is weightu[2].
* weight on partial t is 1 + weightu[2].
* One or more of weights have freedom. Enter your values
by typing ‘weightu[var] = val’ or ‘weight[var] = val’ and
putting ‘;’ between your entries: weightu[2]=1/2;
2
This is the form of rho: c[1] u[1][x, t] + c[2] u[2][x, t]
.
.
.
In this case we gave the weight of u2 via the line weightu[2]=1/2;.
Question 4.8: Do you allow negative weights or fractional weights?
We do not allow negative weights. If one of the weights is determined and turns out to be
negative, the program aborts all computations. Fractional weights are allowed.
Question 4.9: Is there any limit on the number of evolution equations in the system?
Theoretically, there is no limit on the number of equations. In practice, the computations may
take a very long time and may require a lot of memory. The speed depends largely on the
physical amount of memory in the computer.
Question 4.10: Can I compute conserved densities for any type of system of evolution equations?
No, the evolution equations must have polynomial terms in the dependent variables (no integral
terms either). Furthermore, the terms should not depend on x and t explicitly. Thus far, only
x and t are allowed as independent variables.
Question 4.11: Does the program compute non-polynomial conserved densities?
No, the program only computes polynomial type conserved densities. More precisely, only
polynomials in the dependent variables and their derivatives can be obtained.
Question 4.12: How can I get the latest version of the program?
The software (condens.m and the data files) is available by anonymous FTP from
cartan.mines.edu. Login with anonymous. Use your email address or name as password. The
software is in the directory pub/software/mathematica/condens.
æ
5. OTHER SOFTWARE PACKAGES
We will now briefly review other people’s work related to the development of symbolic software
for the computation of conserved densities. In the first section, we give the results from comparing other software packages with ours. In the second section, we address ongoing projects
and software development for conserved densities. Since these projects are still in progress,
only partial information is available. In most cases, no code has been released yet.
5.1 Existing Software Packages
5.1.1 SYMCD
SYMCD was written by Ito [16] in 1994. The program is an improved version of CONSD by
Ito and Kako [17], available since 1985. Both of these programs are in REDUCE.
Similar to our program, SYMCD uses scaling properties of the equations, and computes
polynomial-type conserved densities for systems of any number of evolution equations with
uniform rank. CONSD had a limit on the number of evolution equations and this limitation
has been removed in SYMCD. Evolution equations must be polynomial in the dependent
variables and their derivatives, and variables with negative weight are not allowed. While
testing SYMCD, we used the version released in February 1996, with REDUCE 3.5 on an
IBM Risc 6000 workstation. Our test cases included the following equations:
1. Korteweg-de Vries Equation (3.23),
2. 3-Component Korteweg-de Vries Equation (3.42),
3. Fifth-Order Korteweg-de Vries Equation (3.1),
4. Hirota-Satsuma System (3.22),
5. Boussinesq Equation (3.41).
From the tests, we conclude that SYMCD gives the same conserved densities as the ones
obtained with our program (up to terms that can be converted via integration by parts) when
there are no parameters in the system. In such cases, we acknowledge that SYMCD runs faster
than our program. However, when there are parameters in the system, some differences occur.
SYMCD does not seem to handle parameters properly. It produces the necessary conditions
on the parameters for a density of fixed rank to exist, but then it stops. These conditions have
to be analyzed separately. However, after the analysis, these conditions sometimes lead to a
density and sometimes not. After consulting with Dr. Ito, we know why certain conditions
do not lead to conservation laws. In solving the system for the undetermined coefficients,
5. OTHER SOFTWARE PACKAGES
54
SYMCD considers all possible branches in the solution, irrespective of whether or not those
branches could lead to a conserved density. Having noticed that, SYMCD is not as helpful as
our program for systems with parameters.
Another major difference is that parameters with weight are not allowed in SYMCD, which
means that with SYMCD there is no chance to test systems of evolution equations with
non-uniform rank. Below we give the output of SYMCD for (3.23), with density of rank 14:
1: load_package symcd$
2: % Example: KdV Equation
2: u1T := -u1(0)*u1(1)-u1(3)$
3: DRANK(u1T)$
Du1/DT =
- (u1(3) + u1(1)*u1(0))
WEIGHT OF DX = 1
WEIGHT OF DT = 3
WEIGHT OF u1 = 2
RANK OF u1T = 5
4: CDX(u1T,14)$
+++ CONSERVED DENSITY OF RANK( 14 ) +++
CD!#(1) = 648*u1(9)*u1(1) + 2376*u1(7)*u1(1)*u1(0)
2
+ 3564*u1(5)*u1(1)*u1(0) + 14256*u1(4)*u1(3)*u1(1)
3
- 5544*u1(3)*u1(2)*u1(1)*u1(0) + 2772*u1(3)*u1(1)*u1(0)
2
2
2
2
+ 20988*u1(2) *u1(1) + 4851*u1(2)*u1(1) *u1(0)
2
4
7
+ 1155*u1(1) *u1(0) - 11*u1(0)
Further information can be obtained from Dr. Masaaki Ito by sending email to
ito@puramis.amath.hiroshima-u.ac.jp.
5.1.2 DELiA
The PC package DELiA for “Differential Equations with Lie Approach” [4], developed in
the period 1989-1991, is an outgrowth of the SCoLar project by Bocharov and Bronstein,
5. OTHER SOFTWARE PACKAGES
55
completed in 1989. DELiA, written in Turbo PASCAL by Bocharov and his collaborators,
is a commercial computer algebra system for investigating differential equations using Lie’s
approach. The program deals with symmetries, conservation laws, integrability, and equivalence
problems. It has a special routine for systems of evolution equations, which we used in the
computation of conserved densities. We tested DELiA 1.5.1 on an IBM compatible PC with
an Intel 386 processor and 2 MB of RAM. Our test cases included:
1. Korteweg-de Vries Equation (3.23),
2. Fifth-Order Korteweg-de Vries Equation (3.1),
3. Fifth-Order Korteweg-de Vries Equation (Lax Case) with free α,
4. Hirota-Satsuma System (3.22),
5. Hirota-Satsuma System with α = 1/2,
6. Boussinesq Equation (3.41),
7. Dispersiveless Long Wave System (3.36),
8. 3-Component Korteweg-de Vries Equation (3.42).
We conclude that DELiA computes the same densities as the ones listed in Chapter 3 for
(3.23) and (3.22) with α = 1/2, up to terms that differ only via integration by parts. With
DELiA, we were not able to compute densities for (3.41), (3.36), and (3.42). The program
flashed a message on the screen, indicating that these systems were out of its class. Another
drawback of DELiA became obvious when we tried systems with parameters. In this case,
the program cannot compute the densities with the (necessary) conditions automatically. You
have to use the integrability test first, which determines the conditions based on the existence
of formal symmetries. In fact, these conditions are neither necessary nor sufficient. You have
to analyze these conditions manually, and once you have fixed the parameters you can try to
compute densities. For example, we give the output of DELiA for (3.22) with α = 1/2 :
Your equation is vector evolution.
U01(1) = 1/2*U3(1)-6*U(2)*U1(2)+3*U(1)*U1(1);
U01(2) = -U3(2)-3*U(1)*U1(2);
RHO(1) = U(1);
RHO(2) = 3*U(2)^2-3/2*U(1)^2;
RHO(3) = -5*U1(2)^2+5/4*U1(1)^2+5*U(1)*U(2)^2-5/2*U(1)^3;
RHO(4) = 7*U2(2)^2+7/8*U2(1)^2-21*U(1)*U1(2)^2+7*U(2)*U1(1)*U1(2)
-35/4*U(1)*U1(1)^2+21/2*U(2)^4-21/2*U(1)^2*U(2)^2+35/8*U(1)^4;
Further information about DELiA can be obtained from Dr. Alexei Bocharov at alexie@wri.com.
5. OTHER SOFTWARE PACKAGES
56
5.1.3 FS
The program FS for “formal symmetries” was written by Gerdt and Zharkov in REDUCE
[7]. FS can be applied only to polynomial-nonlinear PDEs of evolution type which are linear
with respect to the highest order spatial derivatives, and with non-degenerated, constant and
diagonal coefficient matrix of the highest derivatives. We tested FS with REDUCE 3.5 on an
IBM Risc 6000 workstation for the following systems:
1. Korteweg-de Vries Equation (3.23),
2. Fifth-Order Korteweg-de Vries Equation (3.1),
3. Boussinesq Equation (3.41),
4. Dispersiveless Long Wave System (3.36),
5. 3-Component Korteweg-de Vries Equation (3.42).
We were not able to compute the densities for the systems (3.41), (3.36), and (3.42). On the
other hand, FS computes the same densities as the ones obtained with our program (up to
terms that differ only via integration by parts) for (3.23). Like DELiA, for equations with
parameters FS computes the conditions on the parameters using symmetry approach. For
(3.1) FS and CONDENS.M give the same conclusions about parameters. Below we give the
output of FS, with density of order 5 for (3.23). Note that the density given in the output is
in the Lisp form.
1: load_package fs$
2: lisp procedure kdv()$
2: algebraic begin;
2:
write "******** Korteweg - de Vries equation
2:
write f(1) := u(3,1) + u(1,1)*u(0,1) $
2: end$
3: lisp$
4: end$
4: kdv();
********
Korteweg - de Vries equation
*********
f(1) := u(3,1) + u(1,1)*u(0,1)
nil
5: res(5);
(((((u 1 1) . 2) . -15) (((u 0 1) . 3) . 5)) . 54)
*********";
5. OTHER SOFTWARE PACKAGES
57
Further information can be obtained from Dr. Viladimir Gerdt at
gerdt@jinr.dubna.su.
5.2 Software under Development
Above we discussed software packages for conserved densities that have been released. Now we
address ongoing projects and development of software for conserved densities.
Sanders, Roelofs, and Wang at RIACA and Free University of Amsterdam are developing a
software package in Maple and FORM to compute conserved densities [34, 35, 36]. Their
approach is much more abstract than ours and it relies heavily on concepts from operator
theory (kernels and images). In fact, they use an extension of the total derivative operator to
a Heisenberg algebra which allows them to invert the total derivative on its image. Further
information can be obtained from Dr. Jan Sanders (jansa@can.nl).
Wolf at University of London is designing a package in REDUCE for the computation of conserved densities [45]. His approach is based on solving over-determined systems of PDEs. Such
differential systems show up in the computation of Lie-point symmetries and generalized symmetries. For a review and additional references to Wolf’s work see [9]. For further information,
contact Dr. Thomas Wolf at T.Wolf@maths.qmw.ac.uk.
Ahner, Tschantz, and Cook at Vanderbilt University are working on a similar project in Mathematica [2]. Currently, no information is available about their algorithm or its implementation.
For information contact Dr. John F. Ahner via email to ahnerjf@ctrvax.vanderbilt.edu. æ
6. CONCLUSION
We presented an algorithm for the symbolic computation of polynomial-type conserved densities
for systems of nonlinear evolution equations. The algorithm is implemented in Mathematica
as CONDENS.M. The program CONDENS.M has been tested on many equations from
soliton theory.
Since it is conjectured that a nonlinear PDE with a large (in principle, infinite) number of
conserved densities is completely integrable, our program can be used to test integrability of
systems of evolution equations. Moreover, for systems with parameters, our program allows
one to determine the values of the parameters so that the system will admit conserved densities
of a given rank.
The design of other algorithms applicable to broader classes of equations (such as wave equations, or equations with variable coefficients), algorithms for non-local conservation laws, or
conservation laws with variable coefficients would be interesting topics for further research.
æ
BIBLIOGRAPHY
[1] M.J. Ablowitz and P.A. Clarkson, Solitons, Nonlinear Evolution Equations and Inverse
Scattering, London Mathematical Society Lecture Note Series 149, Cambridge University
Press, London (1991).
[2] J.F. Ahner, Private Communication (1995).
[3] M. Antonowicz and A.P. Fordy, Physica 28D, 345-357 (1987).
[4] A.V. Bocharov, DELiA: a system for exact analysis of Differential Equations using S. Lie
Approach, DELiA 1.5.1 User Guide, Beaver Soft Programming Team, New York (1991).
[5] R.K. Dodd and J.D. Gibbon, Proc. R. Soc. Lond. A 358, 287-296 (1977).
[6] A. Fordy and J. Gibbons, Phys. Lett. 75A, 325 (1980).
[7] V.P. Gerdt and A.Y. Zharkov, Proc. ISSAC ’90, Tokyo, Japan, 250-254 (1990).
[8] R. Herman, American Scientist 80, 350-361 (1992).
[9] W. Hereman, CRC Handbook of Lie Group Analysis of Differential Equations, Volume
3: New Trends in Theoretical Developments and Computational Methods, Chapter 13,
367-413. Ed.: N.H. Ibragimov, CRC Press, Boca Raton, Florida (1995).
[10] W. Hereman and W. Zhuang, Acta Applicandae Mathematicae 39, 361-378 (1995). Also:
Proceedings of KdV ’95 Conference, April 1995, Amsterdam, The Netherlands. Eds.: M.
Hazewinkel, H.W. Capel and E.M. de Jager, Kluwer Academic Publishers, Dordrecht, The
Netherlands, 361-378 (1995).
[11] F.J. Hickernell, Stud. Appl. Math. 69, 23-49 (1983).
[12] R. Hirota and J. Satsuma, Phys. Lett. A 85, 407-408 (1981).
[13] R. Hirota and M. Ito, J. Phys. Soc. Jpn. 52, 744-748 (1983).
[14] M. Ito, J. Phys. Soc. Jpn. 49, 771-778 (1980).
[15] M. Ito, Phys. Lett. A 91, 335-338 (1982).
[16] M. Ito, Comput. Phys. Commun. 79, 547-554 (1994).
[17] M. Ito and F. Kako, Comput. Phys. Commun. 38, 415-419 (1985).
[18] R.I. Joseph and R. Egri, Phys. Lett. 61A, 429-430 (1977).
Bibliography
60
[19] D.J. Kaup, Studies in Appl. Math. 62, 189-216 (1980).
[20] S. Kawamoto, J. Phys. Soc. Japan 53, 2465-2471 (1984).
[21] M.D. Kruskal, R.M. Miura, C.S. Gardner and N.J. Zabusky, J. Math. Phys. 11, 952-960
(1970).
[22] B.A. Kupershmidt, J. Phys. A: Math. Gen. 18, 571-573 (1985).
[23] B.A. Kupershmidt, Mechanics Research Comm. 13, 47-51 (1986).
[24] B.A. Kupershmidt and G. Wilson, Invent. Math. 62, 403-436 (1981).
[25] R.J. LeVeque, Numerical Methods for Conservation Laws, Lectures in Math., Birkhäuser
Verlag, Basel (1992).
[26] D. Levi, A. Sym and S. Wojciechowski, J. Phys. A: Math. Gen. 16, 2423-2432 (1983).
[27] P.S. Lomdahl, Los Alamos Sci. Spring 10, 27-31 (1984).
[28] S.V. Manakov, Sov. Phys. JETP 38, 248-253 (1974).
[29] R.M. Miura, J. Math. Phys. 9, 1202-1204 (1968).
[30] R.M. Miura, SIAM Rev. 18, 412-459 (1976).
[31] R. Miura, C.S. Gardner and M.D. Kruskal, J. Math. Phys. 9, 1204-1209 (1968).
[32] A.C. Newell, J. Appl. Mech. 50, 1127-1137 (1983).
[33] P.J. Olver, Applications of Lie Groups to Differential Equations, Graduate Texts in Math.,
Springer-Verlag, Berlin (1986).
[34] J.A. Sanders and M. Roelofs, Technical Report 2, RIACA, Amsterdam (1994).
[35] J.A. Sanders, M. Roelofs, and J.P. Wang, Technical Report, RIACA, Amsterdam (1995).
[36] J.A. Sanders, J.P. Wang, Technical Report WS-441, Vrije Universiteit, Amsterdam
(1995).
[37] J.M. Sanz-Serna, J. Comput. Phys. 47, 199-210 (1982).
[38] J. Satsuma and D.J. Kaup, J. Phys. Soc. Japan 43, 692-697 (1977).
[39] K. Sawada and T. Kotera, Prog. Theor. Phys. 51, 1355-1367 (1974).
[40] A.C. Scott, F.Y. Chu and D.W. McLaughlin, Proc. IEEE 61, 1443-1483 (1973).
[41] F. Verheest and W. Hereman, Physica Scripta 50, 611-614 (1995).
[42] R. Weinstock, Calculus of Variations, McGraw-Hill Book Company Inc., New York (1952).
[43] G.B. Whitham, Linear and Nonlinear Waves, Wiley & Sons Inc., New York (1974).
Bibliography
61
[44] R. Willox, W. Hereman and F. Verheest, Physica Scripta 52, 21-26 (1995).
[45] T. Wolf, Private Communication (1995).
[46] S. Wolfram, Mathematica, Addison-Wesley Publishing Company, Inc., Massachusetts
(1992).
[47] V.E. Zakharov (Ed.), What is Integrability?, Springer Series in Nonlinear Dynamics,
Springer-Verlag, Berlin (1990).
[48] V.E. Zakharov and A.B. Shabat, Sov. Phys. JETP 34, 62-69 (1972).
æ