COMPUTATION OF SCALING INVARIANT LAX
PAIRS WITH APPLICATIONS TO
CONSERVATION LAWS
by
Jacob Rezac
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:
Jacob Rezac
Signed:
Dr. Willy Hereman
Thesis Advisor
Golden, Colorado
Date
Signed:
Dr. Willy Hereman
Professor and Interim Department Head
Department of Applied Mathematics and Statistics
ii
ABSTRACT
There has been a large amount of research on methods for solving nonlinear partial differential equations (PDEs) since the 1950s. A completely integrable nonlinear
PDE is one which admits solutions, given specific constraints. These integrable nonlinear PDEs can be associated with a system of linear PDEs through a compatibility
condition. Such a system is called a Lax pair. While Lax pairs are very important in
the theory of solving nonlinear PDEs, few methods exist to compute them. This Thesis presents a method for generating Lax pairs for a specific class of nonlinear PDEs.
Conservation laws, well-known from physics, are a related concept. Indeed, the existence of an infinite number of conservation laws for a nonlinear PDE also predicts
it complete integrability. We discuss a method for the computation of conservation
laws, given a PDE’s Lax pair. This method is based on work done by Drinfel’d and
Sokolov in 1985, which does not seem to have been fully explored in literature. The
goal of this Thesis is to demonstrate the efficacy of these two construction methods.
As such, we also list a number of Lax pairs and conservation laws computed by the
methods presented in the Thesis.
iii
TABLE OF CONTENTS
ABSTRACT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii
LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vi
ACKNOWLEDGMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii
CHAPTER 1 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
CHAPTER 2 PRELIMINARIES . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.1
Lax Pairs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.2
AKNS Scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.3
Gauge Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.4
Conservation Laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
CHAPTER 3 CONSTRUCTION OF LAX PAIRS . . . . . . . . . . . . . . . . 25
3.1
Scaling Invariance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
3.2
Exhaustive Lax Pair Computation Methods . . . . . . . . . . . . . . . 28
3.3
Uniform Weight Construction . . . . . . . . . . . . . . . . . . . . . . . 34
3.4
Weak Lax Pairs and Triviality Concerns . . . . . . . . . . . . . . . . . 39
CHAPTER 4 THE CONSTRUCTION OF CONSERVATION LAWS FROM
LAX PAIRS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
4.1
The Drinfel’d Sokolov Method for Computing Conservation Laws . . . 45
4.2
Triviality and Simplification . . . . . . . . . . . . . . . . . . . . . . . . 51
CHAPTER 5 FURTHER EXAMPLES . . . . . . . . . . . . . . . . . . . . . . 55
5.1
Lax Pairs Computed with Exhaustive Methods . . . . . . . . . . . . . . 55
iv
5.2
5.3
5.1.1
Modified Korteweg-de Vries Equation . . . . . . . . . . . . . . . 55
5.1.2
Kaup-Kuperschmidt Equation . . . . . . . . . . . . . . . . . . . 56
5.1.3
Lax 5th-Order Equation . . . . . . . . . . . . . . . . . . . . . . 57
5.1.4
Sawada-Kotera Equation . . . . . . . . . . . . . . . . . . . . . . 57
5.1.5
Harry-Dym Equation . . . . . . . . . . . . . . . . . . . . . . . . 58
5.1.6
Drinfel’d-Sokolov System . . . . . . . . . . . . . . . . . . . . . . 59
Lax Pairs Computed by Uniform Weight Method . . . . . . . . . . . . 59
5.2.1
Nonlinear Schrödinger Equation . . . . . . . . . . . . . . . . . . 59
5.2.2
Gardner Equation . . . . . . . . . . . . . . . . . . . . . . . . . . 60
Conservation Laws Produced by the Drinfel’d-Sokolov Method . . . . . 61
5.3.1
Korteweg-de Vries Equation . . . . . . . . . . . . . . . . . . . . 61
5.3.2
Modified Korteweg-de Vries Equation . . . . . . . . . . . . . . . 61
5.3.3
Sine-Gordon Equation . . . . . . . . . . . . . . . . . . . . . . . 62
5.3.4
Sinh-Gordon Equation . . . . . . . . . . . . . . . . . . . . . . . 63
5.3.5
Nonlinear Schrödinger Equation . . . . . . . . . . . . . . . . . . 63
CHAPTER 6 CONCLUSIONS AND FURTHER RESEARCH . . . . . . . . . 67
REFERENCES CITED . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
APPENDIX -
OUTLINE OF DRINFEL’D-SOKOLOV ALGORITHM
AND USE OF MATHEMATICA . . . . . . . . . . . . . . . . . 73
v
LIST OF TABLES
Table 5.1
The first four conservation laws for the KdV equation (2.4).
Table 5.2
The first four conservation laws for the mKdV equation (2.31).
Table 5.3
The first four conservation laws for the sine-Gordon equation (5.4).
63
Table 5.4
The first four conservation laws for the sinh-Gordon equation (5.5).
64
Table 5.5
The first four conservation laws for the NLS equation (5.2).
vi
. . . . 61
. . 62
. . . . 65
ACKNOWLEDGMENTS
Thanks to my Thesis advisor, Willy Hereman, for his countless hours of advice
(mathematical and otherwise), for his friendship, and for constantly pushing me to
be a better mathematician.
Thanks to the Department of Applied Math and Statistics at the Colorado School
of Mines, including all the faculty and staff who have helped me during my time here.
In particular, thanks to my Thesis committee Drs. Paul Martin and Luis Tenorio,
both of whom have taught me more math than I care to admit. Also thanks to Dr.
Mark Hickman, whose brief sabbatical in Golden put the following work on a surer
mathematical footing than could have occurred without him.
Further thanks to the many students in my research group the last three years,
Terry Bridgman, Jennifer Larue, Tony McCollom, Janeen Neri, Sara Clifton, Oscar
Aguilar, Jon Tran, and Allen Voltz. Their work ethic, insightful ideas, and pointed
questions all helped push this Thesis to completion. This research was partially
funded by NSF research award no. CCF-0830783.
Finally, thanks to my family and friends, who both distracted and encouraged me
enough to complete this document, whether they realize it or not. Also to Golden’s
numerous fine coffee shops (in particular Higher Grounds), which kept me caffeinated
and gave me a place to work whenever I needed one.
vii
CHAPTER 1
INTRODUCTION
In the early 1950s, physicists Fermi, Pasta, and Ulam (FPU) investigated [1] an
apparent paradox related to the field of statistical mechanics1 . The research group
was attempting to model wave motion in a nonlinear crystal lattice. Scientific understanding at the time predicted that energy put into the system would become equally
partitioned across the crystal lattice. In an unexpected turn of events, however, numerical experiments showed that nearly all of the energy put into the crystal would
remain in its original position; e.g., if the energy were initially put into the lowest
vibrational mode of a wave, all but 3% of the energy would remain in that same
mode [1].
This problem plagued physicists for almost 10 years. In 1965, however, Zabusky
and Kruskal [3] explored the problem in its continuous limit. Numerical experiments
conducted by Zabusky and Kruskal revealed surprising interactions between solutions
of the continuous FPU model: solitary wave solutions passed through each other,
emerging with no changes in velocity, amplitude, or shape. Although the problem was
nonlinear in nature, wave solutions interacted as though they were linear and obeyed
the superposition principle. This discovery hinted at a solution to the FPU paradox.
In the continuous limit, these non-interacting waves were analogous to energy not
mixing in different wave modes of a nonlinear lattice. Zabusky and Kruskal called
these waves solitons2 to emphasize both the solitary nature of the waves and the
similarities these waves shared with particles (e.g., the electron or positron).
1
In recent years [2] it has been argued that Tsingou should also be credited with these investigations. Most of the numerical computations presented in the paper attributed to FPU were, in
fact, done by Tsingou.
2
Zabusky originally planned to call these waves solitrons. Solitron, however, was already the
name of a manufacturing company (which still exists today) [4].
1
It was soon noticed that the equation studied by Zabusky and Kruskal was, in
fact, a nonlinear partial differential equation (PDE) discovered much earlier in the
context of water waves. This so-called Korteweg-de Vries (KdV) equation was derived
in 1895, by Dutch mathematicians Korteweg and de Vries, to describe solitary wave
behavior observed by naval engineer John Scott Russell in 1834 [5]. Soon, other
equations with soliton solutions were found. The modified KdV (mKdV) for example,
which is related to the KdV equation by a simple transformation, was investigated
by the Zabusky-Kruskal team [3], as well as Miura [6], in the mid-’60s. The sineGordon equation, which had been known by plasma physicists for some time, was
soon discovered [7] to have soliton solutions as well. As more and more equations of
this type were uncovered, the study of exact solutions to nonlinear PDEs exhibiting
soliton behavior quickly became a subject important to mathematicians.
In a series of papers [6, 8] published in the 1960s and ‘70s, Gardner, Greene,
Kruskal, and Miura (GGKM) found analytic solutions to describe the interaction
of waves governed by the KdV equation. Importantly, GGKM developed a general
method to find these solutions. The papers also describe other important analytic
properties of the KdV equation and its solutions, such as conservation laws.
In 1968, Lax proposed a formal technique [9] for finding soliton solutions to nonlinear evolution equations based on work done by GGKM. His technique involved
relating the original nonlinear PDE to two linear operators via a compatibility condition. These linear operators, called a Lax pair, are the main topic of this Thesis. The
work done by Lax was further generalized by Zakharov and Shabat [10] and Ablowitz,
Kaup, Newell, and Segur (AKNS) [11]. Soliton solutions were found for more complicated nonlinear PDEs, such as the Nonlinear Schrödinger equation (NLS), and Lax
pairs were again required for the solutions. The solution technique created by these
researchers is called the Inverse Scattering Transform (IST). It has been very successful over the last 40 years, and has been called one of the most important techniques
2
developed in applied mathematics over that time.
During the time-period Lax pairs were first being discovered, mathematicians began to examine conditions under which soliton waves could occur. Many nonlinear
waves which had been studied before the discovery of solitons had exhibited “shock
wave” solutions, requiring jump boundary conditions and conservation law formulations. Applying this analogy to the new nonlinear waves, Miura [6], along with
Kruskal, Zabusky, and Whitham [9], discovered nine conservation laws related to
the KdV equation in the late 1960s. The first few conservation laws correspond to
physical conserved quantities of the system - conservation of mass, momentum, and
energy [12]. The rest, however, have no obvious physical meaning. More and more
conservation laws were soon found, and it was proven that an infinite number existed.
It was soon found that most PDEs having the property of infinite conservation laws
also have a Lax pair and analytic soliton solutions [13]. Conservation laws, and their
relationship to Lax pairs, are another subject discussed in this Thesis.
The connections between exact solutions to a PDE and the existence of either
Lax pairs or an infinite number of conservation laws are well-studied [13]. In the
years since these properties were first discovered, there have been many attempts to
algorithmically compute Lax pairs and conservation laws for a given PDE. In particular, the computation of conservation laws has been successful [12, 14]. However, the
algorithmic construction of Lax pairs has received much less attention [15]. This Thesis will outline algorithmic methods for the computation of both. The methods will
require stringent properties for the PDEs being examined, but many of the equations
important in nonlinear physics have these properties.
The Thesis will proceed as follows: Chapter 2 will discuss the technical aspects
of Lax pairs, conservation laws, and a related concept, gauge transformations. This
Chapter will cover the theory and give concrete examples to further clarify the concepts. Chapter 3 will present two related methods for the computation of Lax pairs.
3
These methods are loosely based on past research, but have not appeared in literature. Particular attention is paid to a type of triviality often ignored by those
researching Lax pairs [16]. Chapter 4 will discuss the computation of conservation
laws from Lax pairs. Specifically, it will discuss a method based on work by Drinfel’d
and Sokolov [17] from which conservation laws can be constructed algorithmically.
Finally, Chapter 5 will list a number of Lax pairs and conservation laws constructed
by the methods described in Chapters 3 and 4. Many of the Lax pairs are more general than those published in literature. The conservation laws listed in this chapter
are compared to published results.
4
CHAPTER 2
PRELIMINARIES
The attempts in the 1960s and ’70s to create a general theory for solving nonlinear PDEs led to the discovery of a number of properties related to exactly solvable
equations. Some of these properties will be discussed in this chapter and a number
of examples will be given to further elucidate concepts.
2.1
Lax Pairs
Perhaps the most basic nonlinear PDE studied in applied mathematics is Burgers’
equation [18],
ut + uux − κuxx = 0.
(2.1)
Here, and throughout this document, u(x, t) is a dependent variable, and x and t
are spatial and temporal independent variables, respectively. Moreover, subscripts
denote partial derivatives. Equation (2.1) developed as a simplification of the NavierStokes equations [19] has applications ranging from fluid flow to optical tomography
and traffic flow [20, 21]. A solution to (2.1) can be found by the so called Hopf-Cole3
transformation [23, 24], which linearizes (2.1) into the (linear) heat equation,
H(v) = vt − κvxx = 0.
(2.2)
In particular, the change of variables governed by
u = −2κ
vx
∂ ln v
= −2κ
∂x
v
(2.3)
reduces (2.1) to
2κ
v2
∂
vx − v
H(v) = 0.
∂x
3
While Hopf and Cole independently discovered this transformation in the early 1950s [18], it
appears as though Florin found the same solution first while studying properties of soil mechanics
as early as 1948 [22].
5
ln v
That is, if v solves (2.2), then u = −2κ ∂∂x
solves (2.1).
This Hopf-Cole solution to Burgers’ equation suggests a technique for solving
nonlinear PDEs:
1. Find a transformation which changes a given nonlinear PDE into a linear one.
2. Solve the new linear equation.
3. Transform back to find solutions to the original nonlinear problem.
This is exactly what the IST attempts to do, though in a much more complicated
way. Of particular interest to the rest of this Thesis is step 1 - finding transformations.
This process of finding transformations such as (2.3) will be generalized and made
algorithmic in the sections that follow.
To develop a linearization technique for nonlinear PDEs, we consider the KdV
equation [5],
ut + 6uux + uxxx = 0,
(2.4)
for u(x, t). Note that the 6 in front of the uux -term can be scaled to any constant.
The similarity between (2.4) and (2.1) suggests a linearizing transformation akin to
the Hopf-Cole transform, (2.3). Indeed, consider the transformation
u=−
ψxx
,
ψ
for a new dependent variable ψ(x, t). Rewriting yields
ψxx + uψ = 0.
(2.5)
Moreover, the KdV equation is invariant under a Galilean transformation [7], u →
u + κ, for some constant κ. So, for some constant λ, we can write (2.5) as
ψxx − (λ − u)ψ = 0.
(2.6)
Equation (2.6) is a well-known Sturm-Liouville equation, related to the Schrödinger
2
∂
equation of quantum physics [7]. We then define the linear operator L = ∂x
2 + uI
by
6
Lψ = λψ.
(2.7)
Note that I in the definition of L is the identity operator. While this sort of
transformation specified the solution for Burgers’ equation, it is unlikely to do so for
more complicated PDEs. To allow for additional flexibility, also consider an operator
M to describe the time evolution of the eigenfunction ψ,
M ψ = ψt .
(2.8)
Assuming4 λt = 0, consider the t-derivative of (2.7),
Lt ψ + Lψt = λψt .
Substituting in from (2.7) and (2.8) gives that
Lt ψ + L(M ψ) = λ (M ψ) = M (λψ) = M Lψ,
or
(Lt + [L, M ])ψ = 0,
where [L, M ] = LM −M L is the commutator of L and M . Continuing, we can readily
verify [15] that
M ψ = −4ψx3 − 6uψx − 3ux ψ
fits the role of (2.8).
The preceding process suggests a general formulation for finding linear PDEs
compatible with a nonlinear PDE. Let a pair of operators L and M be such that
Lψ = λψ and M ψ = ψt and, consequently,
(Lt + [L, M ])ψ =
˙ 0,
(2.9)
where =
˙ means equal on solutions of the nonlinear PDE. Then L and M are known
as a scalar operator Lax pair of the PDE. Moreover, (2.9) is called the operator Lax
4
This assumption is related to solution theory regarding the linear Schrödinger equation and is
required [5] for the IST to work.
7
equation.
Example 1
Substituting the pair of operators
L=
∂2
+ uI
∂x2
and
M = −4
∂3
∂
− 3ux I.
− 6u
3
∂x
∂x
into the Lax equation, (2.9), yields
(Lt + [L, M ]) ψ = −(ut + 6uux + uxxx )ψ.
So, on solutions of the KdV equation (2.4), we see that the operators L and M are a
Lax pair compatible with the KdV equation.
The concept of the Lax pair was expanded further by both the Zakharov-Shabat
team and the AKNS research group in the 1970s. Rather than searching for L and
M operators consistent with a PDE, they considered n × n-matrices. In particular,
they considered matrices X = X(λ, u, ux , uxx , . . .) and T = T (λ, u, ux , uxx , . . .) such
that
XΦ = Φx
and
T Φ = Φt ,
(2.10)
where Φ is a vector of eigenfunctions. Indeed, as X and T depended on the spectral
parameter λ, AKNS and Zakharov and Shabat considered a more general scheme
than that considered in the operator formalism discussed above. Note that, again,
we required λt = 0. Following a similar technique as above, a compatibility condition
relating X, T , and the original nonlinear PDE can be derived. Requiring Φxt = Φtx
gives that
Φxt = Xt Φ + XΦt = Tx Φ + T Φx = Φtx .
Simplifying this and replacing Φt and Φx with (2.10) gives
(Xt − Tx + [X, T ])Φ = 0,
where [X, T ] is the standard commutator,
[X, T ] = XT − T X.
8
Thus, we take
Xt − Tx + [X, T ] =
˙ 0
(2.11)
to be the compatibility condition for a matrix Lax pair X and T . Here, 0 means
the n × n zero-matrix (in the remainder of this Thesis, the distinction between the
zero matrix and zero scalar will be clear from context). Equation (2.11) is called
the matrix form Lax equation (or zero-curvature condition). Note that trivial pairs
are avoided again by requiring compatibility with solutions of the PDE. The matrix
formulation of the Lax equation will be used more commonly throughout the Thesis,
so we say “Lax pair” to mean matrix form of the Lax pair.
Example 2 Consider again the KdV equation, (2.4). Moreover, let Φ = [ψ
φ]T
be a vector-valued function such that ψ and φ are functions of u, ux , and higherorder x−derivatives of u. This type of dependence is said to be on the jet space of
u. Then, we want to find a matrix X which satisfies XΦ = Φx and a matrix T
such that T Φ = Φt . If such X and T exist, there must be a functional relationship
between ψ, φ, ψx , and φx . So, we make a simple guess5 that φ = ψx . We assume φ
and ψ are sufficiently smooth so that φx = ψxx . Furthermore, we know from (2.6)
that ψxx = λψ − uψ. Thus, we can replace ψxx with lower-order terms, giving
φx = λψ − uψ. Thus, there is enough information to construct a matrix X,
0
1
X=
,
λ−u 0
which satisfies Φx = XΦ.
Finding the T matrix is not as immediate as finding X. The process, however,
is the same. Assume again that φ = ψx . Then, again assuming that φ and ψ
are sufficiently smooth, φt = (ψx )t = (ψt )x . However, we know that ψt = M ψ =
−4ψxxx − 6uψx − 3ux ψ. So,
5
This guess is motivated by the way second-order ordinary differential equations can be recast
into matrix form.
9
φt = (−4ψxxx − 6uψx − 3ux ψ)x .
This can be further reduced with the relations ψx = φ, ψxx = λψ − uψ, and ψxxx =
λψx − ux ψ − uψx , yielding
φt = (−4λ2 + 2λu + 2u2 + uxx )ψ − ux φ.
Similarly,
ψt = −4ψxxx − 6uψx − 3ux ψ
= −4(λφ − uφ − ux ψ) − 6uφ − 3ux ψ
= ux ψ − (4λ + 2u)φ.
We have accomplished the goal of this example: φt and ψt are expressed in terms
of φ and ψ after all x-derivatives of these functions have been removed. Thus, since
Φt = [ψt
φt ]T , we have computed the matrix T as
ux
−4λ − 2u
T =
.
−4λ2 + 2λu + 2u2 + uxx
−ux
A direct computation shows that these X and T satisfy the matrix Lax pair equation
along with the KdV equation, (2.4). Indeed, after some algebra, substituting X and
T into (2.11) gives
Xt − Tx + [X, T ] = −
0
0
ut + 6uux + uxxx 0
,
and substitution of (2.4) will give the zero-matrix.
The above matrices can be generalized to generic α in the KdV equation,
ut + αuux + uxxx = 0.
(2.12)
In this case,
X=
0
1
1
λ − 6 αu 0
and
10
,
(2.13)
T =
−4λ2 +
1
αux
6
1
1 2 2
αλu + 18
α u
3
+ 16 αuxx
−4λ − 13 αu
− 61 αux
.
(2.14)
Notice that substituting α = 6 into (2.13) and (2.14) generates the matrices found in
Example 2, as expected.
2.2
AKNS Scheme
Finding a Lax pair for the KdV equation was somewhat intuitive. Other equations,
however, do not produce compatible pairs so easily. Of particular interest, then, is
work done by the AKNS group to construct integrable equations from classes of
compatible Lax pairs. They took a reverse route: rather than constructing a pair
for a given equation, these researchers started from general Lax pairs and tried to fit
them to specific equations.
In the simplified version of the AKNS method, only 2 × 2 matrices are considered
(for a review of higher-dimensional matrices, see [25]). We will do the same. So,
begin by assuming a general form of Lax matrices such that
−iκ q
X=
,
r iκ
(2.15)
where κ is an eigenvalue and q(x, t) and r(x, t) are undetermined functions. The form
of X comes from a generalized approach to the IST taken in [10]. For reasons which
will become clear in section 2.3, we take κ to be the spectral parameter, rather than
λ. Moreover, consider a very general T matrix,
A B
T =
,
C D
(2.16)
where A, B, C and D are general functions of κ, and the jet space of u. Then, keeping
in mind that the Lax equation is equivalent to XΦ = Φx and T Φ = Φt , where Φ = [ψ
φ]T , we have
ψx = −iκψ + qφ,
φx = rψ + iκφ,
and
ψt = Aψ + Bφ,
φt = Cψ + Dφ.
11
(2.17)
(2.18)
If we require symmetry of second-derivatives, ψxt = ψtx and φxt = φtx , and that
κt = 0, we find
qt φ + qφt − iκψt = Ax ψ + Aψx + Bx φ + Bφx
rt ψ + rψt + iκφt = Cx ψ + Cψx + Dx φ + Dφx .
Replacing ψx , φx , ψt , and φt from their definition in (2.17) and simplifying yields
(Ax − iAκ + rB)ψ + (qA + Bx + iBκ)φ = (qC − iAκ)ψ + (qt − iBκ + qD)φ
(Cx − iCκ + rD)φ + (qC + Dx + iDκ)φ = (rt + rA + iCκ)ψ + (rB + iDκ)φ.
So, requiring that φ and ψ be non-zero, we find the system of equations
Ax
Bx + 2iκB
Cx − 2iκC
−Dx
= qC − rB,
= qt − (A − D)q,
= rt + (A − D)r,
= qC − rB.
We can reduce this system of differential equations, without loss of generality [5],
by taking D = −A. While there are a number of ways to find solutions to this system,
we consider expanding A, B, and C into power series in κ (other authors e.g., [26]
expand A, B, and C differently). In doing so, certain evolution equations will be
found to hold on q or r, which will, in turn, determine the PDEs with which the Lax
pairs are compatible. An example clarifies this procedure.
Example 3 Expand A, B, and C up to the third power of κ,
A = A0 + A1 κ + A2 κ2 + A3 κ3 ,
B = B0 + B1 κ + B2 κ2 + B3 κ3 ,
C = C0 + C1 κ + C2 κ2 + C3 κ3 ,
12
where Ai , Bi and Ci are functions on the jet space of q and r. While the reason
behind expanding to the third order in κ is not clear a priori, we will find that this
expansion produces useful results6 . Then, substituting this expansion into (2.11) and
simplifying, gives
0 = (rB0 − qC0 + (A0 )x ) + (rB1 − qC1 + (A1 )x ) κ + (rB2 − qC2 + (A2 )x ) κ2
+ (rB3 − qC3 + (A3 )x ) κ3 ,
qt = (2qA0 + (B0 )x ) + (2(iB0 + qA1 ) + (B1 )x ) κ + (2(iB1 + qA2 ) + (B2 )x ) κ2
+ (2(iB2 + qA3 ) + (B3 )x ) κ3 + 2iB3 κ4 ,
−rt = (2rA0 − (C0 )x ) + (2(iC0 + rA1 ) − (C1 )x ) κ + (2(iC1 + rA2 ) − (C2 )x ) κ2
+ (2(iC2 + rA3 ) − (C3 )x ) κ3 + 2iC3 κ4 .
We hope to solve for A, B, C, q, and r which specify a Lax pair associated with a
PDE. So, splitting these equations in powers of κ yields
κ0
κ1
κ2
κ3
κ4
:
:
:
:
:
rB0 − qC0 + (A0 )x
rB1 − qC1 + (A1 )x
rB2 − qC2 + (A2 )x
rB3 − qC3 + (A3 )x
2iB3 = 0,
= 0,
qt = 2qA0 + (B0 )x ,
−rt = 2rA0 − (C0 )x ,
= 0, 2(iB0 + qA1 ) + (B1 )x = 0, 2(iC0 + rA1 ) − (C1 )x = 0,
= 0, 2(iB1 + qA2 ) + (B2 )x = 0, 2(iC1 + rA2 ) − (C2 )x = 0,
= 0, 2(iB2 + qA3 ) + (B3 )x = 0, 2(iC2 + rA3 ) − (C3 )x = 0,
2iC3 = 0.
From the equations at order κ4 , we see that B3 = C3 = 0. Substituting B3 = C3 = 0
into the equations at order κ3 will give A3,x = 0, or A3 = a3 , where a3 is some
constant. Hence, the equations at order k 3 reduce to
iB2 + a3 q = 0
and
iC2 + a3 r = 0.
Consequently, B2 = ia3 q and C2 = ia3 r. Substituting these into the equations at
order κ2 gives A2,x = 0, or A2 = a2 , for some constant a2 . Then,
ia3 qx + 2(iB1 + a2 q) = 0
and
ia3 rx − 2(iC1 + a2 r) = 0.
Thus, B1 = ia2 q − 21 a3 qx and C1 = ia2 r + 12 a3 rx .
6
In general, there is a lack of motivation in the order of power series expansion. This is one
drawback to the AKNS method.
13
We continue by solving the equations at order κ1 . These determine that (A1 )x =
1
a (qrx
2 3
+ qx r), so that A1 =
1
2
(a3 qr + a1 ), for some constant a1 . Then, from the
remaining equations at order κ1 ,
1
1
2
2iB0 + a1 q + ia2 qx + a3 q r − qxx = 0,
2
2
and
1
1
2
2iC0 + a1 r − ia2 rx + a3 qr − rxx = 0.
2
2
Solving gives
1
B0 =
2
1
2
ia1 q − a2 qx + ia3 (q r − qxx ) ,
2
1
C0 =
2
1
2
ia1 r + a2 rx + ia3 (qr − rxx ) .
2
and
Finally, from the equations at order κ0 , we see
1
1
(A0 )x =
a2 (qx r + qrx ) + ia3 (qxx r − qrxx ) ,
2
4
so
1
1
A0 = a0 + a2 qr + ia3 (qx r − qrx ) ,
2
4
where a0 is some constant. Moreover, the remaining equations at order κ0 specify q
and r as evolution equations. Specifically,
1
1
1
1
2
qt = 2a0 q + ia1 qx + a2 q r + qxx + ia3 3qqx r − qxxx ,
2
2
2
2
(2.19)
1
1
1
2
rt = −2a0 r + ia1 rx + a2 qr + rxx + ia3 (3qrrx − rxxx ) .
2
2
2
(2.20)
and
Substituting a0 = a1 = a2 = 0, a3 = −4i, and r = −1 into (2.19) and (2.20)
yields
14
qt + 6qqx + qxxx = 0,
the KdV equation. The goal of this exercise was to represent T as a cubic polynomial
on the jet space of u. From the above method, we have found q, r, A, B, and C
which form a valid Lax pair when substituted into (2.15) and (2.16). Thus, we have
constructed a Lax pair for the KdV equation,
−iκ q
XAKNS =
,
−1 iκ
(2.21)
and
TAKNS =
−4iκ3 + 2iqκ − qx 4qκ2 + 2iqx κ − 2q 2 − qxx
−4κ2 + 2q
4iκ3 − 2iqκ + qx
.
(2.22)
Substituting this back into the Lax equation (2.11) yields
0 qt + 6uux + uxxx
(XAKNS )t − (TAKNS )x + [XAKNS , TAKNS ] =
,
0
0
so the pair is indeed valid.
A number of other Lax pairs can be found this way. Important examples include
the NLS equation, which comes from expanding A, B and C as second-order polynomials and the sine-Gordon equation, which comes from expanding in inverse powers
of κ. For a larger list of these Lax pairs, as well as a more detailed version of the
above discussion, see [5].
The AKNS scheme was successful in helping to classify integrable systems. It also
broadened the form of Lax pairs mathematicians were willing to use. However, the
whole approach had one major flaw: if a researcher had an equation which needed
to be solved, the scheme would not necessarily produce a Lax pair for it. Attempts
to fix the problem require much creativity on the part of a researcher, and were not
guaranteed to work [27]. Moreover, some require techniques and language difficult for
non-experts to interpret [28]. The methods described in Chapter 3 are an attempt
to fix this. Before considering new computation techniques, however, we discuss two
15
more important properties of integrable PDEs.
2.3
Gauge Transformations
At first glance, the above sections seem to be surprising, if not contradictory. In
particular, two separate Lax pairs have been given for the same equation. Given that
both of these pairs are valid, some questions arise: is there a relationship between
these two pairs? Is one pair better than the other? If so, can one pair easily be transformed into the more useful pair? The first of these questions will be addressed by the
following section on gauge transformations. The other questions will be addressed in
the following Chapters.
Since we know that more than one matrix Lax pair exists for the KdV equation,
we hypothesize that this is true in general. In particular, we look at the vector Φ used
in the definition of X and T , (2.10); if a different Φ, say Φ̂ is used, it seems likely
that a different X and T , say X̂ and T̂ will be required. Thus, consider a nonsingular
matrix G such that Φ̂ = GΦ, where Φ = [ψ
φ]T , as before. Then, we can relate X
and X̂ by noting that
Φ̂ = G (Φ)x + Gx Φ = (GX + Gx )Φ = (GX + Gx )G−1 Φ̂ = X̂ Φ̂.
x
Rewriting this, we see that
X̂ Φ̂ = (GXG−1 + Gx G−1 )Φ.
(2.23)
T̂ Φ̂ = (GT G−1 + Gt G−1 )Φ.
(2.24)
Similarly, we find that
Thus, for any pair of matrices which satisfy (2.11), an infinite number of equivalent
pairs may be found through a gauge transformation of the form (2.23)-(2.24).
In general, constructing a gauge between two given Lax pairs (or, more generally,
from one known Lax pair to a pair of a given form) is nontrivial. In the simplest
cases, however these gauge transformations are fairly straightforward to create.
16
Example 4 Consider again the two Lax pairs found for the KdV equation. Let
X=
0
1
λ−u 0
,
T =
ux
−4λ − 2u
−4λ2 + 2uλ + uxx + 2u2
−ux
,
and
X̂ =
−iκ u
−1 iκ
,
T̂ =
−4iκ3 + 2iuκ − ux 4uκ2 + 2iux κ − 2u2 − uxx
−4κ2 + 2u
4iκ3 − 2iuκ + ux
.
Rewriting equations (2.23) and (2.24), we need to find a nonsingular matrix G
such that
X̂G = GX + Gx
and
T̂ G = GT + Gt .
Because both X and X̂ are linear in u, let us assume that G is a matrix with only
constant entries g11 , g12 , g21 , g22 ,
G=
g11 g12
g21 g22
.
(2.25)
Since Gx = Gt = 0, this significantly reduces the gauge relationship. So, we have
−ig11 κ − g21 u −ig12 κ + g22 u
X̂G =
,
ig21 κ − g11
ig22 κ − g12
and
GX =
g12 λ − g12 u g11
g22 λ − g22 u g21
.
These must match element-wise, which yields
g11 = −ig12 κ + g22 u,
g21 = ig22 κ − g12 ,
ig11 κ − g21 u = −g12 λ + g12 u,
ig21 κ − g11 = g22 λ − g22 u.
Substituting g11 and g21 from (2.26)-(2.27) into (2.28)-(2.29) yields
17
(2.26)
(2.27)
(2.28)
(2.29)
g12 (κ2 + λ) = 0,
g22 (κ2 + λ) = 0.
If we set λ = −κ2 , this is satisfied. Thus, if G is nonsingular, we have that
−ig12 κ + g22 u g12
G=
,
(2.30)
ig22 κ − g12
g22
and λ = −κ2 , will relate X and X̂. Moreover,
2
2
u,
− 2ig12 g22 κ + g22
det(G) = g12
so as long as g12 or g22 is nonzero, G will be a legitimate gauge transformation.
However, g22 multiplies a u term in the (1, 1)-element of (2.30). Since we assumed
that G was a constant matrix, we must set g22 = 0. This yields
−ig12 κ g12
G=
.
−g12
0
To match the gauge transformation found in literature [15], take g12 = 1. This
gives
G=
−iκ 1
−1 0
.
In fact, using this G gives
−1
GXG
=
−iκ −κ2 − λ + u
−1
iκ
.
Substituting λ = −κ2 gives X̂, as expected.
More complicated gauge transformations can also be found. In the next example,
we consider a more complicated case, for which a constant gauge matrix cannot be
found. Rather than turning (2.23) and (2.24) into a simple algebraic system, we will
be forced to work with a system of differential equations.
Example 5 Consider the mKdV equation [6],
ut + αu2 ux + uxxx = 0.
18
Common values for α are ±6 and ±1. In this example, we will consider the case
α = −6,
ut − 6u2 ux + uxxx = 0.
(2.31)
Consider the following two Lax pairs from the mKdV equation,
0
1
X=
λ + u2 + ux 0
(2.32)
with
T =
−2uux − uxx
−4λ + 2u2 + 2ux
.
2uux + uxx
−4λ2 − 2λu2 − 2λux + 2u4 + 4u2 ux − 2uuxx − uxxx
(2.33)
This pair can be computed in the same way as were X1 and T1 from Example 4. A
second pair [5] is
X̂ =
−ik u
u ik
(2.34)
with
T̂ =
−4ik 3 − 2iku2
4k 2 u + 2ikux + 2u3 − uxx
2
3
4k u − 2ikux + 2u − uxx
4ik 3 + 2iku2
,
(2.35)
obtained through the AKNS-scheme. Now, we want to find a nonsingular matrix G
such that X̂ = GXG−1 + Gx G−1 . We first naively consider a constant gauge matrix,
as in Example 4,
G=
g11 g12
g21 g22
.
Equating terms element-wise as before, we get
g11 = −ig12 κ + g22 u,
g21 = ig22 κ + g12 u,
g12 (λ + u2 + ux ) = −ig11 κ + g21 u,
g22 (λ + u2 + ux ) = ig21 κ + g11 u.
19
Substituting g11 and g21 from the first two equations in the above system into the last
two equations gives
g12 (λ + ux ) = −g12 κ2 ,
g22 (λ + ux ) = −g22 κ2 .
Assuming ux is non-constant, the only solution is g12 = g22 = 0. However, this would
also mean that g11 = g21 = 0, leading to G = 0. Thus, the choice of a constant matrix
is too limited, and must be modified.
Because the sets of Lax pairs have u2 as their highest order, we assume G is linear
in u and its first derivative,
G=
g11 (u, ux ) g12 (u, ux )
g21 (u, ux ) g22 (u, ux )
,
(2.36)
for arbitrary functions g11 , g12 , g21 , and g22 . Then, substituting G into (2.23) gives
the system of equations
X̂G − GX − Gx = 0.
Substituting X and X̂ gives 4 equations in terms of g11 , g12 , g21 , g22 , u, and spectral
parameters. Specifically, the following must be satisfied
ig12 κ − g22 u + g12,x + g11 = 0,
ig22 κ + g12 u − g22,x − g21 = 0,
(2.37)
(2.38)
ig11 κ − g21 u + g11,x + g12 λ + u2 + ux = 0,
ig21 κ + g11 u − g21,x − g22 λ + u2 + ux = 0.
(2.39)
(2.40)
Assume for the moment that g12 and g22 are non-constant. Say, e.g., they are
functions of u. Then, (2.37) and (2.38) require that g11 and g21 be quadratic functions
of u. However, this violates the assumption that G is linear in u. Thus, we assume
that g12 and g22 are constants, say g12 = α and g22 = β. Substituting these values
into (2.39) and (2.40) yields
20
ig11 κ − g21 u + g11,x + α(λ + u2 + ux ) = 0,
ig21 κ + g11 u − g21,x − β(λ + u2 + ux ) = 0,
(2.41)
(2.42)
while (2.37) and (2.38) give that g11 = βu − iαk and g21 = αu + iβk. Now replacing
these values of g11 and g21 into the remaining equations and simplifying gives
α(κ2 + λ) + (β + α)ux = 0,
β(κ2 + λ) + (β + α)ux = 0.
One solution to this system is λ = −k 2 and (α + β) = 0. That is, α = −β. Thus, the
system (2.37)-(2.40) is satisfied if
g12 = −g22 = α,
g11 = −α(iκ + u),
g21 = α(−iκ + u),
and
λ = −κ2 .
Substituting the above into G gives
−α(iκ + u) α
G=
.
α(−iκ + u) −α
For α = −1, this simplifies into
2.4
ik + u −1
ik − u 1
.
Conservation Laws
The concept of a conservation law is well-known from physics. Conservation of
mass, energy, and momentum are common phrases even among non-specialists. While
this Thesis focuses primarily on Lax pairs, conservations law are also discussed. We
introduce the concept now, in anticipation of what will follow in Chapters 4 and 5.
In particular, a conservation law for a given PDE is an equation of the form
ρt + Jx =
˙ 0.
21
(2.43)
The function ρ(x, t) is called a conserved density and the function J(x, t) is the
associated flux. Both ρ and J are functions on the jet space of u. As before, =
˙
means equality on solutions of the PDE.
Earlier we saw that a PDE with a Lax pair could be solved exactly with the IST.
Similarly, the existence of an infinite number of conservation laws predicts integrability7 [13]. Moreover, conservation laws can be used to find other properties of interest
for a PDE. For example, the Miura transformation, which helped lead to the discovery
of a Lax pair for the KdV equation, was discovered [7] in the context of conservation
laws. Furthermore, the knowledge of a conservation law is often helpful for finding
a numerical solution to a PDE. Thus, finding conserved quantities is an important
topic. In section 4, we will discuss a construction technique based on Lax pairs. In
this section, however, we will demonstrate the concept of conservation laws and their
relationship to Lax pairs with an example.
Example 6
Consider again the KdV equation, (2.4). It is well-known that there
are an infinite amount of conservation laws related to the KdV equation [6]. The first
two conservation laws are simple:
(u)t + 3u2 + uxx
(u2 )t + 4u3 − u2x + 2uuxx
x
x
=
˙ 0,
=
˙ 0.
These are related to conservation of mass and momentum, respectively. In view of
both, we might expect a pattern to form, with the next density being ρ = u3 . So,
given this density, we try to find a flux, J, by solving
u3
t
+ (J)x =
˙ − 3u2 (6uux + uxxx ) + (J)x = 0.
That is,
(J)x = 3u2 (6uux + uxxx ) .
7
The converse is not always the case. The Burgers’ equation, for example, can be solved exactly
through linearization, but only has a finite number of conservation laws [29].
22
However, there is no way to express a J (on the jet space of u) which satisfies the
above equation. So, the guess of ρ = u3 was wrong. In fact, a nontrivial computation
[9] shows that
1 2
9 4
1 2
3
2
2
u − ux +
u − 6uux + 3u uxx + uxx − ux uxxx =
˙ 0.
2
2
2
t
t
So, we see that even in simple cases, conservation laws are non-trivial to construct.
Many methods, however, have been developed with this construction in mind (for
a survey of these methods, see, e.g., [12] or [14]) One of these methods, developed
by Zakharov and Faddeev [10] in 1971, is related to a PDE’s Lax pair. We will
demonstrate their method by algorithmically developing the above conservation laws
for the KdV equation.
Example 7
Recall that for the KdV equation, (2.4), the Lax operator L (2.6) was
given by,
Lψ − λψ = ψxx − (λ − u)ψ = 0.
Making the substitution ψ = eiκx+σ(x,t) turns this into
eiκx+σ 2iκσx + u + (σx )2 + σxx = 0,
where we have used the substitution λ = −κ2 . Assuming eiκx+σ 6= 0, this reduces to
a Riccati equation in terms of σx ,
2iκσx + u + (σx )2 + σxx = 0.
Similarly, from
ψt = Aψ + Bψx ,
the same substitution and simplification gives
σt = A − (iκ + σx ) B.
(2.44)
Note that, in Example 2, we set up our Lax pair this way, setting A = ux and
B = 4κ2 − 2u. So, using standard techniques for solving Riccati equations [5], we
23
expand µ = σx as
µ=
∞
X
n=1
µn
,
(2iκ)n
where µn are functions on the jet space of u. Substituting this into (7) will give [5]
µ1 = −u,
µ2 = u x ,
µ3 = −(u2 + uxx ),
...
µn+1 = (−1)n+1
n−1
X
!
µp µn−p + (µn )x ,
n ≥ 2.
p=1
Note that (µk )x = µn−1 , for odd n. So, even numbered µi s are total derivatives
and we consider them to be trivial. A more rigorous definition of triviality will be
given in Chapter 4. Using A = ux and B = 4κ2 − 2u and substituting the above
expansion for µ into (2.44), we get
"
"∞
#
!#
∞
X
X µn
µ
n
= ux − 4κ2 − 2u
+ iκ
.
n
n
(2iκ)
(2iκ)
n=1
n=1
t
(2.45)
x
Equating terms in negative odd powers of κ will give a listing of the conservation laws
for the KdV equation. Note that this example requires the existence of a Lax pair.
However, there are equations with a finite number of conservation laws but without
a Lax pair. For a review of equations of that type, see, e.g., [12].
24
CHAPTER 3
CONSTRUCTION OF LAX PAIRS
It is common, in differential equation theory, to use symmetry arguments to find
characteristics of a given differential equation. Lie-point symmetries, for example,
can be used to generate solutions to a differential equation from an already-known
solution [30]. Other types of symmetry, specifically scaling or (dilation) symmetry,
have been used to construct conservation laws [12] and Lax pairs in operator form [15].
We will use this type of symmetry to generate Lax pairs in matrix form as well.
Recall that a Lax pair is only valid if it satisfies the Lax equation, (2.11), when
evaluated on the PDE. This suggests that properties of a Lax pair are associated
with properties of the PDE. One property held by many integrable PDEs is scaling
symmetry invariance. As such, we will construct candidate Lax pairs which have the
same symmetry properties as the PDE we are examining. This restricts both the
class of PDEs with which we can work and the type of Lax pairs we can construct for
those PDEs. However, we will find Lax pair construction methods based on scaling
invariance to be adept at constructing Lax pairs in this restricted class. In the next
sections, we discuss the concept of scaling invariance, as well as two specific methods
for the construction of Lax pairs.
3.1
Scaling Invariance
Before considering the construction of Lax pairs, we must elaborate on the mean-
ing of scale invariance. Simply, a PDE is scale invariant if applying a scaling transformation returns the original PDE, multiplied by a constant.
Example 8
Consider, again, the KdV equation, (2.4). Let us assume that the
KdV equation is scaling invariant under a symmetry
(x, t, u) → (κ−a x, κ−b t, κc u) = (ξ, τ, υ),
25
where the exponents a, b, and c must be determined. Substituting this transformation
into (2.4), we have
∂u
∂u ∂ 3 u
υ ∂ hυi
∂ hυi
∂ hυi
+ 6u
+
+6 c
+ 3 c
=
∂t
∂x ∂x3
∂t κc
κ ∂x κc
∂x κ
" #
3
1 ∂
∂τ
1 ∂
∂ξ
1 ∂3
∂ξ
= c
υ
+ 6 2c υ
υ
+ c 3 υ
κ ∂τ
∂t
κ ∂ξ
∂x
κ ∂ξ
∂x
h i
∂ t
∂ x
1 ∂
1 ∂
υ
υ
= c
+ 6 2c υ
b
κ ∂τ
∂t κ
κ ∂ξ
∂x κa
" 3 #
1 ∂3
∂ hxi
+ c 3 υ
κ ∂ξ
∂x κa
=
1
∂υ
1 ∂ 3υ
1 ∂υ
+
6
υ
+
.
κc+b ∂τ
κ2c+a ∂ξ κc+3a ∂ξ 3
Thus, for the KdV equation to be scaling invariant, we must have
c + b = 2c + a = c + 3a.
(3.1)
This system of equations is under determined, so we set a = 1 and find b = 3 and
c = 2. Thus, the KdV equation scales like
(x, t, u) → (κ−1 x, κ−3 t, κ2 u) = (ξ, τ, υ).
Furthermore, every monomial of the equation scales with a κ−5 . Because of this, the
KdV equation is scaling invariant.
The calculations required to find scaling symmetries were by no means complicated. Doing the same computations with a computer algebra system, however, is
not so straightforward. To simplify the process of finding scaling symmetries, we introduce the concept of a variable’s weight. Using this tool, we can reduce the process
of finding scaling symmetries to one of solving an algebraic equation such as (3.1).
In doing so, we turn the problem into linear algebra which can be easily done by a
computer program.
Definition 3.1.1 (Weight). The weight of a variable in a PDE, denoted W (·), is
the exponent of κ in the variable’s scaling symmetry. By convention, W (x) = −1,
26
W (constant) = 0.
This definition can be extended to find the weights of a derivative as well. In
∂
= −W (ν). Thus, by convention,
particular, for some dependent variable ν, W ∂ν
∂
W ∂x
= 1. This tool can also be extended to find the weight of more complicated
∂
terms. For example, W (ut ) = W (u) + W ∂t
. With this definition in hand, we
compute the same symmetry as above, but in a much simpler manner.
Example 9
Assume the KdV equation (2.4) is weight invariant. To find the weights
of each variable, we set up a system of algebraic equations by equating the weights
of each monomial in the PDE. Moreover, we require that each term in the PDE has
an equal weight. This gives
∂
∂
∂
W (u) + W
= 2W (u) + W
= W (u) + 3W
.
∂t
∂x
∂x
∂
Note that this is the same system of equations as (3.1), with W ∂x
= −a,
∂
∂
W ∂t
= −b, and W (u) = c. Since, by convention, W ∂x
= 1, we get that
∂
= 3. We found above that (x, t, u) → (κ−1 x, κ−3 t, κ2 u), so
W (u) = 2 and W ∂t
this result is unsurprising. Weights and scaling symmetries will be used extensively
in the construction techniques in the next section. Another important tool in the
construction of Lax pairs is the computation of monomials of a certain weight. If we
assume weights are always nonnegative and integer8 , this process is simple. Consider,
∂
e.g., a PDE such that W (u) = W ∂x
= 1. Then, we can list every combination
on the jet space of u of a certain weight. For example, every weight 1 term is of the
form c1 u, where c1 is some constant. Similarly, every weight 2 term is either of the
form c2 ux or c3 u2 , for arbitrary constants c2 and c3 . No other combination involving
u, ux , uxx , . . . can be of weight 2. The weight approach will prove to be very important
in Lax pair construction techniques.
8
This is not always appropriate. See, e.g., [12], for an example of non-integer weights
27
3.2
Exhaustive Lax Pair Computation Methods
We discuss the computation of Lax pairs with an exhaustive search technique.
Assume that if a PDE has a certain scaling invariance, then that same scale must be
present in Lax pairs and the Lax equation. That is, assume that the Lax pair inherits
the symmetry properties of the PDE. We further assume that each element of X
and T is weight invariant (though we do not initially specify what those weights will
be). To justify this technique, consider the Lax pairs already presented in Chapter 2.
∂
will produce weight-invariant elements in each9 example
Assuming W (λ) = W ∂x
given.
To begin, assume that the X and T matrices are such that
W (X) =
W (X11 ) W (X12 )
W (X21 ) W (X22 )
and
W (T ) =
W (T11 ) W (T12 )
W (T21 ) W (T22 )
, (3.2)
where, e.g., W (X11 ) is the weight of each monomial in the (1, 1) element of X. Then,
we require that each element of the matrix Lax equation is scale invariant. So,
substituting X and T from (3.2) into the Lax equation (2.11) gives the overdetermined
system of equations
W (X11 ) + W
W (X12 ) + W
W (X21 ) + W
W (X22 ) + W
∂
∂t
∂
∂t
∂
∂t
∂
∂t
= W (X12 ) + W (T21 ),
W (T11 ) + W
= W (X12 ) + W (T22 ),
W (T12 ) + W
= W (X22 ) + W (T21 ),
W (T21 ) + W
= W (X22 ) + W (T22 ),
W (T22 ) + W
9
∂
∂x
∂
∂x
∂
∂x
∂
∂x
= W (T11 ) + W (X11 ),
= W (T11 ) + W (X12 ),
= W (T21 ) + W (X11 ),
= W (T21 ) + W (X12 ),
The reader may notice that W (λ) = 2 in Example 1. However, λ does not actually play the role
of the spectral parameter in this example. Indeed, the spectral parameter is κ such that −iκ2 = λ.
Hence, W (κ) = 1, as expected.
28
W (T12 ) + W (X21 ) = W (X11 ) + W (T11 ),
W (T12 ) + W (X21 ) = W (X12 ) + W (T21 ),
W (T12 ) + W (X22 ) = W (X11 ) + W (T12 ),
W (T12 ) + W (X22 ) = W (X12 ) + W (T22 ),
W (T22 ) + W (X21 ) = W (X21 ) + W (T11 ),
W (T22 ) + W (X21 ) = W (X22 ) + W (T21 ),
W (T22 ) + W (X22 ) = W (X21 ) + W (T12 ).
Solving this will give the required weight of each element of the X and T matrices.
Specifically, we have
W (X) =
∂
W ∂x
W (X12)
∂
∂
) − W (X12 ) W ∂x
2W ( ∂x
(3.3)
and
W (T ) =
∂
W
∂t
∂
W ∂t
− W (X12 ) + W
∂
∂x
W
∂
∂t
+ W (X12) − W
∂
W ∂t
∂
∂x
,
(3.4)
where W (X12 ) is undetermined. The reader should compare this with the Lax pairs
listed in, e.g., Example 2 and Example 3.
With the required form of X and T specified, the construction method proceeds
quite quickly. We generate candidate X and T matrices, whose elements have weights
as described above, with undetermined coefficients. Then, substituting X and T
into the Lax equation, we form a system of equations to solve for the undetermined
equations.
Example 10
We make this method concrete by again looking at the KdV equation
∂
= 1, we
(2.4). For this example, set W (X12 ) = 0. Assuming W (λ) = W ∂x
construct candidate Lax pairs of the form
W (X) =
1 0
2 1
,
such that
X=
c1 λ
c2
2
c3 λ + c4 u −c1 λ
29
,
(3.5)
and
W (T ) =
3 4
2 3
,
such that
T =
T11 T12
T21 −T11
,
where
T11 = c5 λ3 + c6 λu + c7 ux ,
T12 = c8 λ2 + c9 u,
T21 = c10 λ4 + c11 λ2 u + c12 λux + c13 u2 + c14 uxx .
Here, ci , i = 1, . . . , 14 are undetermined constants. Note that X and T are assumed
to be traceless, which can be assumed without loss of generality [31]. With this
candidate Lax pair in hand, we substitute (3.5) and (10) back into the Lax equation
(2.11), evaluate on the PDE, and find the coefficients which force the Lax equation
to be zero on the PDE. So, we have four equations to solve, one from each matrix
element. The first equation, for example, is
(c2 c10 − c3 c8 ) λ4 + (c2 c11 − c4 c8 − c3 c9 ) λ2 u + (c2 c13 − c4 c9 ) u2
+ (c2 c12 − c6 ) λux + (c2 c14 − c7 ) uxx = 0.
This equation, along with three others of that type, is difficult to solve. However,
since we have further requirements that λ 6= 0 and u 6= 0, we may split these equations
by powers of λ and on the jet space of u. This reduces the above equation, for example,
to a set of algebraic equations:
c10 c2 = c3 c8 ; c2 c11 = c4 c8 + c3 c9 ; c2 c13 = c4 c9 ; c6 =
c2 c12 ; and c7 = c2 c14 .
Solving these, along with the similar equations from the other three matrix elements of the Lax equation, gives a number of solutions. Most of these solutions,
however, are trivial (e.g., one of the matrices is entirely constant or zero). One of
the more useful solutions is c1 = 0; c2 = − cc210 ; c3 =
11
30
c11
; c4
2
=
c211
; c5
c10
= 0; c6 = 0; c7 =
c2
c2
c2
c10
11
11
1; c8 = −2 c10
; c14 = − c10
; c10 6= 0; and c11 6= 0.
3 ; c9 = 2 c2 ; c12 = 0; c13 = −2 c
10
11
11
Substituting this back into the candidate Lax pair, (3.5) and (10), gives
#
"
0
− cc10
2
11
X = 1 c11
2
(c
λ
+
2c
u)
0
10
11
2 c10
and
T =
T11 T12
T21 −T11
,
where
T11 = ux ,
c10
2
c
λ
−
c
u
,
10
11
c311
c2
= c10 λ4 + c11 λ2 u − 11 2u2 + uxx .
c10
T12 = −2
T21
Specifically, if we choose c10 = 4, c11 = 2i, and rescale the spectral parameter as
λ2 = −iλ, we get the Lax pair in Example 2.
Thus, based on this example, a
method for Lax pair construction could proceed as follows:
1. Find the scaling symmetry of the PDE.
2. Build a candidate Lax pair based on this symmetry, specified by weight matrices
(3.3) and (3.4).
3. Substitute the candidate Lax pair into the Lax equation and solve for undetermined coefficients.
4. Filter out trivial solutions.
5. Substitute the non-trivial solutions into the candidate Lax pair.
This process has only one non-algorithmic aspect: the choice of weights for the
candidate Lax pairs is determined by the (unknown) weight of the (1, 2)-element of
the X matrix. One possible criterion for the selection of this weight will be addressed
in the Section 3.3. However, the method proposed above is nearly algorithmic, and
the selection of weights has a reasonable amount of choice. Moreover, as shown in
Section 5.1, this method works for a large number of PDEs.
31
To further describe the efficacy of the method, we treat the transcendental sineGordon equation. This example will illustrate two important aspects of the computation method. First, it deals with the case of a zero-weighted u, and, as such, a
negatively-weighted
∂
.
∂t
Moreover, the equation is non-polynomial. This, too, will
cause some complications.
Example 11 The sine-Gordon equation [32],
uxt = sin u,
is quite different than the examples above, and those in Section 5.1. Trying to find
the weights of the PDE, for example, is trickier than usual. We have
∂
∂
+W
= W (sin u).
W (u) + W
∂x
∂t
While there are many ways to deal with this weighting [33], one simple way is to
treat the weight of sin u as zero. Similarly, this choice gives that W (u) = 0 Then,
∂
∂
assuming, as always, that W ∂x
= 1 will give that W ∂t
= −1. Each of these - the
zero weight of u and the negative weight of t - are things we have not yet encountered.
As an example of why these new weights cause problems, consider the construction
of a matrix element of weight 1. Certainly, we should have a λ term. However, what
about a λu term? Or a λu2 term? Or even a λ−2 uxxx term? It is all-but impossible
to choose a cutting-off point, at least algorithmically. For this Thesis, we will deal
with this problem somewhat arbitrarily, and further research is warranted.
The other main issue in constructing a candidate Lax pair involves the form of the
PDE. Before, when the PDEs were polynomial on the jet space of u, it made sense
for each matrix element to also be polynomial on the jet space of u. Here, however,
the sin u term makes that assumption invalid. Thus, the candidate pairs will contain
undetermined functions of u, as well as undetermined combinations of polynomial
terms. As we will see, this will require the solving of differential equations as well as
algebraic equations.
32
With these considerations in mind, we assume that each element of the X matrix
has weight 1 and that each element of the T matrix has weight −1. Then, assume
there are functions a and b such that W (a(u)) = W (b(u)) = 0. Based on the known
Lax pair given in [5], we consider the candidate Lax pair10
c1 λ c 2 u x
X=
c3 ux −c1 λ
and
"
T =
a(u)
λ
b(u)
λ
b(u)
λ
− a(u)
λ
#
.
Putting these into the Lax equation and separating by powers of λ and terms on the
jet space of u, we get the system of equations
a0 (u) = (c2 − c3 )b(u),
2c1 b(u) = −c2 sin u,
= c3 sin u,
0
b (u) = −2c2 a(u)
= 2c3 a(u).
One solution to this mixed system of algebraic and differential equations is
c1 = −i,
1
c2 = −c3 = − ,
2
1
a(u) = i cos u,
4
1
and b(u) = i sin u.
4
Hence,
−iλ − 21 ux
1
u
−iλ
2 x
cos u sin u
sin u − cos u
X=
and
1
T =
i
4λ
.
This is exactly the Lax pair given in [5]. Since the candidate Lax pair was based
on that Lax pair, this is expected.
10
This candidate pair can quite easily be extended to hold, say, more polynomial terms in the X
matrix. For the purposes of this example, however, we ignore these terms.
33
The above is a toy example. Other problems with transcendental PDEs, such as
Liouville’s equation [33], do not admit Lax pairs so easily.
3.3
Uniform Weight Construction
The method for Lax pair construction presented in the previous section allows
for Lax pair matrices to have elements of different weights. This is a very general
solution technique and, given a perfect nonlinear algebraic equation solver, would
be sufficient for the construction of scaling invariant Lax pairs. However, due to
computational complexity, even the most advanced solvers cannot handle some basic
examples. In this section, we discuss a more structured method, which requires a
certain weight-invariance in each element of the X matrices and a, possibly different,
weight-invariance in each element of the T matrix.
Consider again the defining weight equations, (3.3)-(3.4). For each element of
∂
= W (X12 ) =
the matrix X to have equal weight, it must be true that W ∂x
∂
∂
2W ( ∂x
) − W (X12 ). Hence, W (X12 ) = W ∂x
. So the overall weight of the X matrix
must be equal to this quantity as well. By a similar argument, it follows that the
weight of the T matrix must be equal to the weight of
∂
.
∂t
Note that this method won’t
work for every PDE. The KdV equation, for example, requires that either W (X12 )
∂
or W (X21 ) be different from W ∂x
. We will see, however, that there are a number
of important PDEs for which this method works, and so we continue describing this
method.
Once the weights of X and T are established, expand both Lax matrices in a finite
power series in λ. That is, set
∂
W ( ∂x
)
X=
X
∂
W ( ∂t
)
X (i) λi
and
i=0
T =
X
j=0
34
T (j) λj
∂
∂
We will assume11 , for the rest of the discussion, that W ∂x
< W ∂t
. Since we have
∂
1, W (T ) = W ∂t∂ 1, and W (λ) = 1, the weights of
assumed that W (X) = W ∂x
each X (i) and T (j) must be changed accordingly. It is not hard to see that W (X (i) ) =
∂
∂
− i 1 and W (T (j) ) = W ∂t
− j 1 will suffice. In this way, we have
W ∂x
expanded X and T in terms of a sum of matrices whose elements all have the same
weight. Thus, we can find the equations which must be satisfied by X (i) and T (j) to
ensure that X and T will satisfy the Lax equation. To find these relationships, we
substitute X and T into the Lax equation and separate in powers of λ,


∂
W ( ∂x
)

∂
W ( ∂t
)


∂
W ( ∂x
)

∂
W ( ∂t
)

 X

 X (j) j 
 X
  X (j) j 
X (i) λi  − 
T λ  +
X (i) λi  
T λ

i=0
j=0
i=0
t

∂
W ( ∂t
)
j=0
x

(3.6)

∂
W ( ∂x
)

 X (j) j   X
T λ 
X (i) λi  =
˙ 0.
−
j=0
i=0
As differential operators are linear, the only difficulty in the expansion comes from
∂
= 1.
products in the commutator term. To simplify this expansion, assume W ∂x
We have taken this assumption to be standard, so we may simplify as such without
loss of generality. As a further simplifying step, recall that a matrix element with
∂
weight zero is simply a constant. Thus, each element of X (1) and T (W ( ∂t )) is a
constant and their derivatives are zero. With these two considerations in mind, we
separate (3.6) by powers of λ to get
h
X
(1)
i
∂
W ( ∂t
(
))
,T
=
˙ 0
1 h
i
X
∂
−k)
W ( ∂t
(k)
(
)
=
˙ 0,
X ,T
and
k=0
which we call the kinematic equations, along with
X (0)
− T (0)
t
(0) (0) +
X ,T
=
˙ 0
x
and
11
This assumption is based on Lax pairs found in literature.
35
(3.7)
− T
(i)
1 h
i
X
(k)
(W ( ∂t∂ )−i−k) =
+
X
,
T
˙ 0
x
i = 1, . . . , W
k=0
∂
∂t
− 1,
which we call the dynamic equations. At this point, the power of this method is
clear: though we have restricted the form of the Lax pair, we have written down a
relationship which must hold true for any Lax pair of this type. In particular, the
kinematic equations (3.7) can be solved quite easily. The solution is then used to
reduce the complexity of the dynamic equations (3.3).
Once the kinematic equation is solved, we find a Lax pair as before. We construct
candidates of each X (i) and T (j) and use the simplified dynamic equations to find the
undetermined coefficients. To demonstrate this method, we will find the Lax pair for
the mKdV equation.
Example 12 Consider the mKdV equation, (2.31). For the purposes of this example, take α = −6 so that the equation is
ut − 6u2 ux + uxxx = 0.
As described by the method above, we first find the weights of the mKdV equation.
Therefore we solve the system of equations
∂
∂
= 3W (u) + W
,
W (u) + W
∂t
∂x
∂
∂
= W (u) + 3W
,
W (u) + W
∂t
∂x
∂
∂
3W (u) + W
= W (u) + 3W
∂x
∂x
for the weights of u,
∂
,
∂x
and
∂
.
∂t
The third of these equations is a direct consequence
∂
of the first two, so the system is under-determined. Thus, set W ∂x
= 1. Then,
∂
the second equation gives W ∂t
= 3, so the first equation gives W (u) = 1. Thus,
∂
we require W (X) = W ∂x
1 = 1 and W (T ) = W ∂t∂ 1 = (3)1 and assume that X
and T take the form X = X (0) + X (1) λ and T = T (0) + T (1) λ + T (2) λ2 + T (3) λ3 . Using
36
(3.7) the kinematic equations are
[X (1) , T (3) ] = 0,
[X
(0)
,T
(3)
] + [X
(1)
,T
(2)
(3.8)
] = 0.
To facilitate the solution of this system of equation, we first assume a form for X (i) ,
for i = 0, 1 and T (j) , for j = 0, . . . , 3, such that X and T are both traceless. Thus,
set
"
X (i) =
(i)
X11
(i)
X21
(i)
X12
(i)
−X11
#
"
T (i) =
and
(i)
T11
(i)
T21
(i)
T12
(i)
−T11
#
.
Solving (3.8) gives a number of possible solutions. After sorting through the trivial
ones, we have
(1)
(0)
X12
(1)
(3)
X11 T12
(1)
X12 =
(3)
T11
(3)
X11 T21
(1)
X21 =
,
(3)
,
T11
(0)
(3)
(3)
(1)
(3)
(2)
(1)
(2)
(3)
(0)
(3)
(3)
(1)
(3)
(2)
(1)
(2)
(3)
X T T + X11 T11 T12 − X11 T11 T12
,
= 11 11 12
2
(3)
T11
X11 T11 T21 + X11 T11 T21 − X11 T11 T21
.
2
(3)
T11
(0)
X21 =
Substituting these relationships back into the candidate for X gives
X11 X12
,
X=
X21 −X11
where
(0)
(1)
X11 = X11 + X11 λ,
(0)
X12 , =
(3)
(3)
(1)
(3)
(2)
(1)
(2)
(3)
X11 T11 T12 + X11 T11 T12 − X11 T11 T12
(3)
(T11 )3
!
(1)
+
(3)
T11
and
37
(3)
X11 T12
!
λ,
(0)
X21 =
(3)
(3)
(1)
(3)
(2)
(1)
(2)
(3)
X11 T11 T21 + X11 T11 T21 − X11 T11 T21
(3)
(T11 )3
!
(1)
+
(3)
X11 T21
!
(3)
λ.
T11
The T matrix remains unchanged. Now that the commutator relationships have
been solved, we further specify the candidate matrices with scale invariance. Specifically, since W (X) = 1, we have
"
X=
1
c24
c1 λ + c2 u
(c1 c4 c7 λ + (c2 c4 c7 − c1 c5 c7 + c1 c4 c8 ) u)
1
c24
(c1 c3 c4 λ + (c2 c3 c4 − c1 c3 c5 + c1 c4 c6 ) u)
−(c1 λ + c2 u)
Furthermore, since W (T ) = (3)1, we have
T11 T12
T =
.
T21 −T11
where
T11 = c4 λ3 + c5 λ2 u + c11 u2 + c12 ux λ + c9 u3 + c10 uux + c13 uxx ,
T12 = c3 λ3 + c6 λ2 u + c16 u2 + c17 ux λ + c14 u3 + c15 uux + c18 uxx ,
T21 = c7 λ3 + c8 λ2 u + c21 u2 + c22 ux λ + c19 u3 + c20 uux + c23 uxx .
Finally, these candidate X and T matrices are substituted back into the matrix
Lax equation and we solve for the undetermined coefficients. Again, there are many
solutions to this, one of which is c2 = 0; c3 = 0; c4 = −4c31 ; c5 = 0; c6 = −2c21 c14 ; c7 =
c2
0; c8 = −8 c141 ; c9 = 0; c10 = 0; c11 = 2c1 ; c12 = 0; c13 = 0; c15 = 0; c16 = 0; c17 =
−c1 c14 ; c18 = − 21 c14 ; c19 = 4 c114 ; c20 = 0; c21 = 0; c22 = 4 cc141 ; and c23 = −2 c114 . Substituting these coefficients into the candidate pairs, we have
c1 λ 21 c14 u
X=
2
u −c1 λ
c14
and
T =
2
c14
−4c31 λ3 + 2c1 λu2
−2c21 c14 λ2 u − c1 c14 λux + c14 u3 − c14 21 uxx
(−4c21 λ2 u + 2c1 λux + 2u3 − uxx )
4c31 λ3 − 2c1 λu2
38
,
#
.
for undetermined c1 and c14 . Particular choices of these coefficients give nicer answers.
For example, setting c1 = −i and c14 = 2 gives
−iλ u
X=
u iλ
and
T =
−4iλ3 − 2iλu2
4λ2 u + 2iλux + 2u3 − uxx
2
3
4λ u − 2iλux + 2u − uxx
4iλ3 + 2iλu2
.
Notice that this is the same Lax pair given in Example 5 with λ replaced by κ.
Comparatively, this method is computationally simple. Unfortunately, only a
handful of equations seem to admit Lax pairs of this type. See Section 5.2 for two
specific examples.
3.4
Weak Lax Pairs and Triviality Concerns
The methods above appear to be promising. One important topic - that of triv-
iality in the computed Lax pairs - still needs to be addressed. In a series of papers
published in the 1990s and early 2000s, Sakovich [34, 35], Marvan [36], and Calogero
and Nucci [16] define the concept of strong and weak Lax (or ‘true and fake’) Lax
pairs. This represents a different type of triviality for Lax pairs than discussed in
Chapter 2. Essentially, a Lax pair is weak if it is gauge equivalent to a matrix without a spectral parameter. Otherwise, it is strong. To demonstrate this concept, we
consider an example of a weak Lax pair given by Marvan for Burgers’ equation, [36].
Example 13
Consider again Burgers’ equation, (2.1),
ut − uxx − κuux = 0.
For this example, take κ = 1. A Lax pair for Burgers’ equation which can be produced
by the methods presented above (or found in literature, e.g., [35, 36]) is
1
λ
λ + 12 u
X=
−λ
2 −λ + 12 u
and
39
1
T =
4
λu
λu + 12 u2 + ux
−λu + 21 u2 + ux
−λu
.
It is straightforward to verify that substituting these X and T into the Lax equation
yields
1
Xt − Tx + [X, T ] =
4
ut − uxx − uux
0
0
ut − uxx − uux
,
which is the zero matrix upon evaluation on the original PDE. In other words, by
triviality methods discussed thus far, this is a valid and non-trivial Lax pair. Now,
consider the gauge matrix
" √
λ+
1
√
G=
λ−
2
√1
λ
√1
λ
√
λ−
√
λ+
√1
λ
√1
λ
#
.
Gauging X and T with equations (2.23) and (2.24) gives
1
1
1 + 12 u
−1
X̂ = GXG =
−1
2 −1 + 12 u
and
−1
T̂ = GT G
1
=
4
u
1 2
u
−
u + ux
2
1 2
u
2
+ u + ux
−u
.
These are exactly the original X and T , but with the spectral parameter removed.
For applications, these weak Lax pairs are useless. For example, when carrying out
the IST to find solutions to a nonlinear PDE, the existence of a spectral parameter
is crucial [5]. Moreover, the method for constructing conservation laws presented in
Chapter 4 will only work if a spectral parameter is present in a Lax pair. In many
ways, the lack of a spectral parameter in a Lax pair takes away any implication of
integrability for a nonlinear PDE12 . The number of weak Lax pairs that are found by
the computation methods described in this chapter is of concern, and a good solution
has yet to be found. On the positive side, there are algorithmic methods for detecting
12
There are equations admitting weak Lax pairs which are, indeed, completely integrable. However, these weak Lax pairs do not lead to solutions via the IST. See [37] for further discussion of
this.
40
if a Lax pair is weak. We will outline these now, following [31] and [34].
Consider a generic evolution equation, ut = h(u, ux , uxx , . . .) with Lax pair X and
def
T . Moreover, define the covariant derivative as ∇M = Mx −[X, M ] for any matrix M
and the matrix X from the Lax pair. Note that this is not related to the ∇ operator
from Calculus. We use, however, it to be consistent with [35]. Furthermore, define
the matrix C as
def
C = Xu − ∇ (Xux ) + ∇2 (Xuxx ) − . . . .
Here, e.g., ∇2 M = ∇(∇(M )), and so on for higher powers of ∇. Then, it can be
shown [34] that the zero-curvature representation of the evolution equation can be
written in terms of C. Because this new representation is dependent on an invariant
PDE, it is no surprise that C has invariant properties of its own. To find this invariance, define the cyclic basis of C as the sequence of linearly independent matrices
C, ∇C, ∇2 C, . . . , ∇n−1 C. Then, the closure equation of the cyclic basis is
∇n C = a0 Ca1 ∇C + . . . + an−1 ∇n−1 C.
The invariance here comes from the constants ai . Every valid Lax pair of a PDE will
have the same ai . Moreover, if one or more of these ai have a spectral parameter
in it, that parameter cannot be removed from the Lax pair - e.g., it is a strong Lax
pair. This is the key idea to filtering out weak Lax pairs. All that is required is the
computation of the closure equation and examination of the constants in it; due to the
invariance of cyclic bases, the existence of an explicit gauge transform which removes
a spectral parameter from a Lax pair it is not required when testing for triviality. We
clarify these concepts with an example.
Example 14
Consider again the weak Lax pair for the Burgers’ equation given in
(2.1), with κ = 1. We first find the first few covariant derivatives for the X matrix,
41
1 0 1
C=
,
4 1 0
1
−1 −1
∇C = λ
,
1
1
4
u
−1 −1
2
∇C= λ
.
1
1
8
The first two of these matrices are linearly independent. The third, however, can be
represented in terms of the first two. Thus, the closure equation of this cyclic basis
is
1
∇2 C = u∇C,
2
with the coefficient on C being 0. Note, importantly, that this closure equation is
free of any spectral parameters. It follows, then, that X and T are weak Lax pairs,
as expected.
We now apply this method to a strong Lax pair for Burgers’ equation. This, again,
is given in [35] (though earlier references can be found). Consider the pair13
λ + 14 u
0
X=
1
− λ + 41 u
and
T =
−2λ2 + 81 u2 + 14 ux
0
1
1 2
2
2λ − 8 u − 41 ux
−2λ + 2 u
.
We compute the covariant derivatives,
1 1 0
,
C=
4 0 −1
0 0
∇C =
,
− 12 0
0
0
2
∇C=
,
−(λ + 14 u) 0
13
Note that the exhaustive search method of Lax pair computation will indeed construct this
pair, though with considerable difficulty and in a large amount of time.
42
and derive the closure equation,
1
∇ C = 2λ + u ∇C,
2
2
where the coefficient on C is 0 again. Because the closure equation necessarily has a
spectral parameter in it, X and T are strong Lax pairs. The above is an important
warning about the efficacy of the computational methods described in this chapter. It
is easy to construct Lax pairs which appear to be nontrivial which are, in reality, weak
pairs. However, with diligence, weak Lax pairs can be avoided, or at least detected.
43
CHAPTER 4
THE CONSTRUCTION OF CONSERVATION LAWS FROM LAX PAIRS
In this chapter, we discuss a method to construct conservation laws based on Lax
pairs. As discussed in Chapter 2, several method exist for algorithmic computation of
conservation laws for nonlinear PDEs. In the following section, we outline one of these
methods, based on work done in the early 1980s [17] by Drinfel’d and Sokolov (DS).
The method takes advantage of the similarity between the equation which defines a
Lax pair, (2.11), and the equation which defines a conservation law, (2.43), to convert
from the former to the latter. This approach, which we name after its discoverers,
is algorithmic and is based on a simple recursive formula. However, it requires a
specific form of Lax pair and will only admit conservation laws for a narrow class
of PDEs. Perhaps for this reason, the method has been, to the author’s knowledge,
largely ignored in literature.
4.1
The Drinfel’d Sokolov Method for Computing Conservation Laws
Consider again the defining equations for conservation laws and Lax pairs,
ρt + Jx =
˙ 0
Xt − Tx + [X, T ] =
˙ 0.
and
Examining these equations, it is clear that if [X, T ] = 0, the Lax equation represents
a conservation law with density ρ = X and flux J = −T . Thus, given a Lax pair, X
and T , the goal of the DS method will be to force the commutator term to vanish.
Before describing how to do this, we put some restrictions on the Lax pair.
Consider a Lax pair of the form14
X = X0 − X 1 λ
14
(4.1)
It seems obvious to try to expand this in higher (or lower) powers of λ. This approach, however,
has yet to prove fruitful.
45
and
T =
m
X
T i λi
i=0
where X0 , an off-diagonal matrix, and Ti are matrices containing functions on the jet
space of u, X1 is a constant diagonal matrix, m is an undetermined integer, and λ is
the spectral parameter.
Example 15
Consider the KdV equation with a Lax pair given by (2.21)-(2.22).
We can decompose the X matrix into the form of (4.1) where
0 u
X0 =
−1 0
(4.2)
and
i 0
0 −i
X1 =
.
(4.3)
With an appropriate Lax pair at hand, we return to the task of annihilating the
commutator term. Recall that the commutator of two diagonal matrices is always
zero. Thus, to make the commutator in (2.11) vanish, we use a diagonalization
technique.
Theorem 4.1.1 (Drinfel’d-Sokolov [17]). Let S be a matrix-valued series of the form
S=I+
∞
X
Γi λ−i ,
(4.4)
i=1
where I is the appropriately-sized identity matrix and Γi are matrices whose elements
are on the jet space of u. Then, we define X̃ = SXS −1 to be of the form
X̃ = Dx − X1 λ +
∞
X
Pi λ−i ,
(4.5)
i=0
where the Pi are diagonal matrices and Dx is a matrix of
∂
∂x
operators.
Equating X̃S = SX in powers of λ gives the following recurrence relation
46
Pn + [Γn+1 , X1 ] = Γn X0 −
Γ0n
−
n−1
X
Pi Γn−i ,
(4.6)
i=0
which allows each Γi and Pi to be solved recursively.
Proof. First, recall Cauchy’s product formula.
Theorem 4.1.2 (Cauchy’s Product Formula [38]). Consider two series with terms
an and bn , which are assumed to converge as n → ∞. Then,
! ∞ !
∞
∞ X
n
X
X
X
an
bn =
ak bn−k .
n=0
n=0
n=0 k=0
Using this, expand both sides of X̃S = SX. Specifically, to use Cauchy’s product,
we must re-index X̃ to
X̃ = Dx − X1 λ + P0 +
∞
X
Pj+1 λ−(j+1)
j=0
and S to
S=I+
∞
X
Γj+1 λ−(j+1) .
j=0
For what follows, we take Γ0 = I. Thus, expanding these power series, we have
X̃S =
Dx − X 1 λ + P 0 +
∞
X
!
Pj+1 λ−(j+1)
I+
∞
X
j=0
= Dx I +
+
∞
X
j=0
+
j=0
∞
X
X1 Γj+1 λ−j + P0
j=0
P0 Γj+1 λ−(j+1) +
∞
X
Γj+1 λ−(j+1)
j=0
Γ0j+1 λ−(j+1) − X1 λ −
j=0
∞
X
!
∞
X
Pj+1 λ−(j+1)
j=0
!
Pj+1 λ−(j+1)
∞
X
!
Γj+1 λ−(j+1) .
j=0
Expanding the product of sums in the last term with Cauchy’s product formula gives
47
X̃S =
∞
X
Γ0j+1 λ−(j+1)
j=0
∞
X
+
− X1 λ −
P0 Γj+1 λ−(j+1) +
j=0
∞
X
X1 Γj+1 λ−j + P0
j=0
∞
X
Pj+1 λ−(j+1)
j=0
j
+
∞ X
X
Pk+1 Γj−k+1 λ−(j+2) .
j=0 k=0
Similarly, we expand SX as
SX =
I+
∞
X
!
−(j+1)
Γj+1 λ
X0 − X1 λ
j=0
= X0 − X1 λ +
∞
X
−(j+1)
Γj+1 X0 λ
−
j=0
∞
X
Γj+1 X1 λ−j .
j=0
Equating these and simplifying, we get
∞
X
Γ0j+1 λ−(j+1)
−
j=0
+
−j
X1 Γj+1 λ
+
j=0
∞
X
−(j+1)
P0 Γj+1 λ
j=0
+
∞
X
∞
X
Γj+1 X1 λ−j
j=0
+
∞
X
Pj+1 λ
−(j+1)
−
j=0
j
∞ X
X
∞
X
Γj+1 X0 λ−(j+1)
j=0
Pk+1 Γj−k+1 λ−(j+2) + P0 − X0
j=0 k=0
= 0.
Setting n = j + 1 and splitting these in powers of λ yields
P 0 = X0
at order λ0 and
Pn + [Γn+1 , X1 ] = Γn X0 −
Γ0n
−
n−1
X
Pk Γn−k
k=0
at order λn . This is exactly the recursion relation given in (4.6).
48
With this theorem in hand, we can find the first few conservation laws for the
KdV equation.
Example 16
It was shown by Drinfel’d and Sokolov [17] that if P0 = X0 then the
elements of each matrix Pi are a conserved density of the given PDE. Thus, continuing
with the KdV equation (2.4), we solve the recurrence relation (4.6) for n = 0, . . . , 3.
First, we let P0 be
0 0
0 0
P0 =
.
Then, the first few Pi matrices are
1
P1 = i
2
1
P2 = −
4
i
P3 =
8
−u 0
0 u
ux 0
0 0
,
,
(u2 + uxx ) 0
0
−u2
.
Thus,, e.g., ρ1 = 21 iu, ρ2 = − 14 ux , ρ3,1 = − 8i u2 , and ρ3,2 = 8i (u2 + uxx ) are valid
densities. Using only the recurrence relationship (4.6), we can create an infinite
number of conserved densities. The corresponding fluxes could be constructed by
the same process, using the matrix T from the Lax pair with the same S matrix.
However, this calculation quickly grows unwieldy due to complicated matrices being
processed. Thus, we take a simpler approach. Specifically, given a ρ, we can take the
integral of (2.43) with respect to x, giving
Z
Z
ρt dx + Jx dx =
˙ 0.
So, rearranging,
49
Z
J =−
(ρi )t dx,
(4.7)
where ut and the x-derivatives of u are replaced from the PDE. Note that this integral
can be difficult to evaluate. For a straightforward way of evaluating integrals of this
type, see [39]
Example 17 Finally, we finish the derivation of conservation laws for the KdV
equation. As an example, we take a non-zero element from the first computed matrix
density,
1
ρ1 = iu
2
and put it into (4.7). This gives
Z
J1 =
Dt
1
iu dx
2
Z
1
iut dx
2
Z
1
= − i 6uux + uxxx dx
2
1
= − i 3u2 + uxx .
2
=
Thus, we have now constructed a ρ1 and J1 such that
(ρ1 )t + (−J1 )x =
˙ 0.
The fluxes for the next two conservation laws follow similarly. We have
3
1
J2 = iuux + iuxxx
2
4
associated with ρ2 = − 14 iux . Also,
1
1
1
J3,1 = − iu3 + iu2x − iuuxx .
2
8
4
associated with ρ3,1 = 8i u2 and
1
1
7
1
J3,2 = − iu3 − iu2x − iuuxx − iuxxxx
2
8
2
8
50
associated with ρ3,2 = 8i (u2 + uxx ).
These conservation laws are valid, but fail to match the conservation laws given
in literature for the KdV equation. In the next section, we discuss the reasons for
this.
4.2
Triviality and Simplification
While the above densities and fluxes are valid, there are two important points
to consider: trivial conservation laws (to be defined precisely below) are constructed
and the conservation laws given in Example 17 do not match exactly with published
results, e.g., [12, 14].
Two types of trivial conservation law are considered in [12]. The first type occurs
when the conservation law equation (2.43) is satisfied even when not evaluated on
solutions to the PDE. A simple example of this is
(uxx )t + (−uxt )x = 0,
which holds for any sufficiently smooth u. For the purposes of this Thesis, we will
ignore this type of trivial conservation law. The second type occurs when all the
terms from the density (or flux) can be moved into the flux (or density). That is, a
conservation law is trivial if either ρ = 0 or J = 0. An example of this is
3
1
1
iux + i
uux + uxxx
=
˙ 0.
4
2
4
t
x
This is trivial because
1
iux
4
=
t
1
iut
4
,
x
so we can rewrite the conservation law as
1
3
1
(0)t + − iut − iuux − iuxxx
=
˙ 0.
4
2
4
x
The second of these examples may look familiar. In fact, it was ρ2 computed in
Example 17. Moreover, in Chapter 2, we came across this listing of conservation
51
laws, ignoring every other density for this reason. Note that we only have triviality
when all the terms can be moved from, e.g., the density into the flux. For example,
consider ρ3,2 =
i
8
(u2 + uxx ). The (uxx )t term is equivalent to (uxt )x , which can be
moved into the flux. However, since the u2 term cannot be moved by similar means,
the conservation law is non-trivial. It is, however, equivalent to ρ3,1 for this reason.
To make the process of detecting these trivial conservation laws easier, we must first
introduce the so-called Euler operator.
Definition 4.2.1. The Euler operator (sometimes called the Euler-Lagrange operator
or variational derivative) for f = f (x, u(M ) (x)), a function of x, u(x), and up to M derivatives of u(x) is [39]
M X
∂f
∂k
,
Lu(x) f =
− k
∂x
∂ukx
k=0
where ukx = ∂ k uxk .
Using this, we state without proof15 a theorem which will aid us in detecting trivial
conservation laws.
Theorem 4.2.1 (Poole [12]). A function f = f (x, u(M ) (x)) is exactly integrable if
and only if Lu(x) f = 0. That is, f = Fx , for some function F if and only if the
Euler-operator applied to f is identically zero.
Using the Euler operator, we can now algorithmically detect a trivial conservation
law of the second type. Specifically, assume some density, ρ, is such that Lu(x) ρ = 0.
(ρ)
Then, ρ = ∂Jx , for some function J (ρ) . But, this means
(ρ)
(ρ)
,
ρt + Jx = Jx t + (J)x = (0)t + J + Jt
x
and so the conservation law is trivial. Thus, we know that if the density of a conservation law is exact, the conservation law is trivial. We will use this result in picking
out trivial conservation laws and removing them from consideration.
15
See [12] for a thorough discussion and proof of this theorem.
52
Finally, we must remove terms from conserved densities which are linear combinations of terms equivalent to other terms up to divergence. That is, elements whose
Euler-operators are equivalent. To illustrate this, we consider an example.
Example 18
Consider the list of terms
{u3 , u2x , uuxx , uxxxx }.
(4.8)
To find divergence-equivalent terms, apply the Euler-operator to each of these terms
separately,
Lu(x) {u3 , u2x , uuxx , uxxxx } = {3u2 , −2uxx , 2uxx , 0}.
Thus, the term uxxxx is exact and can be removed. Similarly, u2x + uuxx is exact
because Lu(x) u2x + Lu(x) uuxx = −2uxx + 2uxx = 0. Thus, we remove the terms with
highest-order derivatives and (4.8) simplifies into
{u3 , u2x }.
The final simplification step involves the normalization of the term with highestdegree exponents. That is, forcing the coefficient in front of the highest-degree term
to be 1.
Example 19
Consider each of the three conservation laws found in Example 17.
The first conservation law is
1 i (u)t + 3u2 + uxx x = 0.
2
After normalization, this becomes
(u)t + 3u2 + uxx
x
.
The second conservation law is
1
i [(ux )t + (6uux + uxxx )x ] = 0.
4
As discussed, this conservation law is trivial, so we remove it. The third conservation
law is
53
1 − i u2 t + 4u3 − u2x + 2uuxx x ,
8
which simplifies to
(u2 )t + 4u3 − u2x + 2uuxx
x
= 0.
See Appendix A for an outline of the symbolic algorithm to simplify conservation
laws. More examples of conservation laws produced by this method are in section
5.3.
54
CHAPTER 5
FURTHER EXAMPLES
In this chapter, we provide Lax pairs and conservation laws computed with the
methods presented in this Thesis. In particular, the first section lists Lax pairs
computed by the exhaustive search method given in Section 3.2. Section 5.2 lists
Lax pairs computed by the uniform weight method given in Section 3.3. The final
section lists conservation laws computed by the Drinfel’d-Sokolov method presented
in Chapter 4.
5.1
Lax Pairs Computed with Exhaustive Methods
We now give Lax pairs constructed by the methods discussed in Section 3.2. For
each equation, weights and working Lax pairs will be given, as well as simplifications
and comparisons to literature.
5.1.1
Modified Korteweg-de Vries Equation
The modified Korteweg-de Vries (mKdV) equation,
ut − 6u2 ux + uxxx = 0,
has the following weights:
∂
= 1,
W
∂x
W (u) = 1,
and
We generate a Lax pair of
c1 λ c2 u
1
u −c1 λ
c2
T11 T12
T21 −T11
X=
and
T =
where
55
,
W
∂
∂t
= 3.
T11 = −4c31 λ3 + 2c1 λu2 ,
T12 = −4c21 c2 λ2 u − 2c1 c2 λux + 2c2 u3 − c2 uxx ,
1
T21 = −
4c21 λ2 u − 2c1 λux − 2u3 + uxx .
c2
This can be simplified to the Lax pair found in Example 12 with c1 = −i and c2 = 1.
5.1.2
Kaup-Kuperschmidt Equation
The Kaup-Kuperschmidt (KK) equation,
1
5
ut + γ 2 u2 ux + γux uxx + γuuxxx + uxxxxx = 0,
5
2
has weights
W
∂
∂x
= 1,
∂
∂t
W (u) = 2, and
W
−25c1 λ −25c2 λ2 −
1 c2 5
γ u
500 c21
= 5.
We generate a Lax pair with
1
X= 2
γ
"
c2
25 c12
#
25c1 λ
and
T =
T11 T12
T21 −T11
,
where
51
1
1
c1 λuxx − γ 2 uux − γuxxx ,
2γ
25
20
5
1
2
c
1 c2
1 c2 5 3
2 2
= c2 λ2 u2 +
c2 λ2 uxx −
γ λuux −
γλuxxx +
γ u
2γ
25 c1
10 c1
12500 c21
1 c2 4 2
9 c2 4
1 c2 3
γ ux +
γ uuxx +
γ uxxxx ,
2
2
625 c1
5000 c1
500 c12
c21
51
2
u +
uxx .
=−
c2
2γ
T11 = c1 λu2 +
T12
T21
It is unknown if this Lax pair has been reported in literature. The pair given in [40]
is related to the above pair, but the two are certainly distinct. The pair constructed
56
in this section, however, is quite different from pair given in [41].
5.1.3
Lax 5th-Order Equation
For the Lax 5th-order equation,
ut +
3 2 2
γ u ux + 2γux uxx + γuuxxx + uxxxxx = 0,
10
with weights of
W
∂
∂x
= 1,
W (u) = 2, and
W
∂
∂t
= 5,
we generate a Lax pair with
1
iλ − 16
c1 u
8γ
−iλ
5c1
T11 T12
T21 −T11
X=
and
T =
,
where
2
1
1
4
T11 = −16iλ5 + iγλ3 u + γλ2 ux − iγλ 3γu2 + 10uxx − γ (3γuux + 5uxxx ) ,
5
5
50
50
1
1
1
T12 = c1 λ4 u − iλ3 ux − λ2 γu2 + 5uxx + iλ (3γuux + 5uxxx )
2
20
40
1
+
3γ 2 u3 + 30γu2x + 40γuuxx + 50uxxxx ,
800
4 1
T21 = −
γ 800λ4 − 40γλ2 u + 3γ 2 u2 + 10γuxx .
125 c1
When c1 = −16, γ = 10, and λ = −iλ, this is equivalent to the Lax pair presented
in [42].
5.1.4
Sawada-Kotera Equation
The Sawada-Kotera (SK) equation,
1
ut + γ 2 u2 ux + γux uxx + γuuxxx + uxxxxx = 0,
5
has
57
W
∂
∂x
= 1,
W (u) = 2, and
W
∂
∂t
= 5.
In this case, we obtain
1
X= 2
γ
"
1 c2
−25c1 λ − 125
(3125λ2 c21 + γ 5 u)
c2
#
1
c2
25 c21
25c1 λ
and
T =
T11 T12
T21 −T11
,
where
1
25c1 λ γu2 + 10uuxx − γ 2 (γuux + 5uxxx ) ,
25γ
1 c2 1 2 2
2
3125c
λ
γu
+
10u
− 250γ 2 c1 λ (γuux + 5uxxx )
=
xx
1
2
3125 c1 γ
4
2 3
2
+ 5γ γ u + 5γux + 15γuuxx + 25uxxxx ,
T11 =
T12
T21 = −
5.1.5
c21
γu2 + 10uxx .
c2 γ
Harry-Dym Equation
The Harry-Dym equation,
ut + u3 uxxx = 0,
with weights
W
∂
∂x
= 1,
W (u) = 1, and
W
∂
∂t
= 6,
admits a Lax pair of
"
X=
c1 λ + 2 c1c3c2 u
1
c5
(c1 c4 λ2 + c1 c3 λu)
− c1 λ + 2 c1c3c2 u
−4 c1 cc22 c5
3
#
and
"
T =
c5 λ6 + 2 c2c3c5 λ5 u c4 λ7 + c3 λ6 u + c2 λ5 u2 −
−4
c2 c25 5
λ
c3
1 c3 3
u uxxx
2 c5
−(c5 λ6 + 2 c2c3c5 λ5 u)
This differs from the Lax pair given in [5].
58
#
.
5.1.6
Drinfel’d-Sokolov System
Consider the Drinfel’d-Sokolov system of equations [29],
ut + 3vvx = 0,
vt + vux + 2uvx + 2vxxx = 0.
(5.1)
In equation (5.1), both u(x, t) and v(x, t) are both dependent variables. While systems of equations have not yet been considered, they are an important and common
examples of complete integrable PDEs. The Drinfel’d-Sokolov system has weights of
W
∂
∂x
= 1,
W (u) = 2,
W (v) = 2,
and
W
∂
∂t
= 3.
The Drinfel’d-Sokolov system admits a Lax pair of
"
#
c1 λ c2 λ2 + 4cc22 (u − v)
1
X=
c2
− c21
−c1 λ
and
"
T =
c1 λv + 21 vx c2 λ2 v +
c2
1 c2
4 c21
(uv − v 2 + c1 vx + 2vxx )
c1 λv − 21 vx
− c21 v
#
.
To the author’s knowledge, no matrix Lax pair for this equation is reported in the
literature.
5.2
Lax Pairs Computed by Uniform Weight Method
This section will list Lax pairs computed by the methods presented in Section 3.2,
Section 3.3.
5.2.1
Nonlinear Schrödinger Equation
The NLS equation,
iut − uxx − 2u2 u∗ = 0,
where u(x, t) is a complex function and
∗
(5.2)
denotes complex conjugate has weights of
59
W
∂
∂x
= 1,
∗
W (u) = W (u ) = 1,
and
W
∂
∂t
= 2.
We generate a Lax pair with
X=
c1 λ c 2 u
1 ∗
u −c1 λ
c2
,
and
T =
−2ic21 λ2 + iuu∗
−2ic1 c2 λu − ic2 ux
1
∗
∗
2ic21 λ2 − iuu∗
(−2ic1 λu + iux )
c2
.
This is exactly the Lax pair reported in [5] with c1 = −i and c2 = 1.
5.2.2
Gardner Equation
Consider the Gardner equation [43],
ut + αuux + α2 ux + u2 ux + uxxx = 0.
(5.3)
A simple transformation [43] relates the Gardner equation (5.3) to the mKdV equation
(2.31). It’s no surprise, then, that similar methods can be used to construct Lax pairs
for each equation. The Gardner equation has has weights of
W
∂
∂x
= 1,
W (u) = 1,
W (α) = 1,
and
W
∂
∂t
= 3.
Assuming an X with uniform weight of 1 and a T with uniform weight of 3 yields
−iλ
− 65 c1 (α + 2u)
X=
5
(α + 2u)
iλ
144c1
and
T =
T11 T12
T21 −T11 ,
where
60
1
T11 = −4iλ3 + iλ(5α2 + 2αu + 2u2 ),
6
1
T12 = c1 −24λ2 (2u + α) − 24iλux + 5α3 + 12α2 u + 6αu2 + 4u3 + 12uxx ,
5
5
T21 = −
−24λ2 (α + 2u) + 24iλux + 5α3 + 12α2 u + 6αu2 + 4u3 + 12uxx ,
864c1
where c1 is a free constants. This Lax pair has the same form (with different constant
coefficients) as (2.34)-(2.35) under the transformation given in [43].
5.3
Conservation Laws Produced by the Drinfel’d-Sokolov Method
We will show conservation laws from four PDEs. These conservation laws were
computed with the DS method discussed in Chapter 4. In this section, we adopt the
notation unx =
5.3.1
∂nu
∂xn
for n > 3.
Korteweg-de Vries Equation
We return to the KdV equation, (2.4), and list additional conservation laws. Using
X0 and X1 , as given in Chapter 4, the first 4 conservation laws are listed in table
Table 5.1. These are comparable to the conservation laws found in Section 2.4. They
can also be compared to those found in [5].
Table 5.1: The first four conservation laws for the KdV equation (2.4).
Order
1
2
3
4
5.3.2
Density(ρ)
u
u2
u3 − 21 u2x
u4 − 2uu2x + 51 u2xx
Flux(J)
3u2 + uxx
4u3 − u2x + 2uuxx
9 4
u − 6uu2x + 3u2 uxx + 21 u2xx − ux uxxx
2
24 5
u − 18u2 u2x + 4u3 uxx + 2u2x uxx +
5
16
uu2xx − 4uux uxxx − 51 u2xxx + 25 uxx uxxxx
5
Modified Korteweg-de Vries Equation
Consider again the mKdV equation, (2.31). The X part of the Lax pair constructed in Chapter 3 can be decomposed as
61
X 0 − X1 λ =
0 u
u 0
−
i 0
0 −i
λ.
The DS method gives the conservation laws in Table Table 5.2
Table 5.2: The first four conservation laws for the mKdV equation (2.31).
Order
1
Density(ρ)
u2
2
u4 − uuxx
Flux(J)
−3u4 − u2x + 2uuxx
− 4u6 + 6u2 u2x + 10u3 uxx − u2xx
+ ux uxxx − uu4x
9
11
− u8 − 45u4 u2x + u4x + 6u5 uxx
2
2
2
2 2
− 22uux uxx − 23u uxx − 14u2 ux uxxx
1
1
1
− 3u3 u4x + uxx u4x − ux u5x + uu6x
2
2
2
u6 + 5u2 u2x + 12 uu4x
3
126 2 4
24 10
u − 112u6 u2x −
u ux
5
5
154 4 2
406 3 2
u ux uxx −
u uxx
+ 8u7 uxx −
5
5
− 22u2x u2xx + 44uu3xx + 28u4 ux uxxx
628
129 3
ux uxxx +
uux uxx uxxx
−
5
5
106 2 2
129 2
+
u uxxx +
uux u4x
5
5
72
6
208 2
+
u uxx u4x + u2 ux u5x + u3 u6x
5
5
5
1
1
1
− uxx u6x + ux u7x − uu8x
5
5
5
−
21 2
uu uxx
5 x
1
14
+ u2 u2xx − uu6x
5
5
u8 + 14u4 u2x +
4
5.3.3
Sine-Gordon Equation
The sine-Gordon equation [5],
uxt − sin u = 0.
62
(5.4)
with matrix X decomposed as
X0 − X1 λ =
0 − 21 ux
1
u
0
2 x
−
i 0
0 −i
λ
admits conservation laws with the DS method. The first four of these are listed in
table Table 5.3. These are equal to the conservation laws given in [33].
Table 5.3: The first four conservation laws for the sine-Gordon equation (5.4).
Order
1
2
3
Density(ρ)
u2x
u4x + 4ux uxxx
u6x − 20u2x u2xx + 8ux u5x
Flux(J)
2 cos(u)
−4 sin(u)uxx
− (8u4x − 32u2x uxx ) sin(u) + 14u4x cos(u)
336
ux u2xx uxxx
5
64
224 2 2
u u + ux u7x
+
5 x xxx
5
16 16u4x uxx + 11u3xx − 12ux uxx uxxx
5
2
− 24ux u4x + 4u6x sin(u)
8
3u6x − 50u2x u2xx − 72u3x uxxx cos(u)
−
5
u8x − 56u4x u2xx +
4
5.3.4
−
Sinh-Gordon Equation
The sinh-Gordon equation [5], a close relative of the sine-Gordon equation, is
uxt − sinh(u) = 0.
(5.5)
The X Lax pair for the sinh-Gordon can be decomposed as [5]
0 21 ux
i 0
X0 − X 1 λ = 1
−
λ.
u
0
0 −i
2 x
The conservation laws given in Table Table 5.4 are equivalent to those given in [33].
5.3.5
Nonlinear Schrödinger Equation
Consider again the NLS equation, (5.2). We can write the X from the Lax pair
of the NLS as [5]
63
Table 5.4: The first four conservation laws for the sinh-Gordon equation (5.5).
Order
1
2
3
4
Density(ρ)
u2x
u4x − 4ux uxxx
u6x + 20u2x u2xx + 8ux u5x
Flux(J)
−2 cosh(u)
4 sinh(u)uxx
−8 (4u2x uxx + u4x ) sinh(u) + 14 cosh(u)u4x
336
ux u2xx uxxx
5
224 2 2
64
+
ux uxxx − ux u7x
5
5
u8x + 56u4x u2xx +
X 0 − X1 λ =
16 −
11u3xx − 16u4x uxx − 12ux uxx uxxx
5
− 24u2x u4x − 4u6x sinh(u)
8
+
3u6x + 50u2x u2xx + 72u3x uxxx cosh(u)
5
0 u
u∗ 0
−
i 0
0 −i
.
The software we developed to compute conservation laws does not accept complex
PDEs. As such, to compute the conservation laws given in Table 5.5, we rewrote
(5.2) as the system
ut − 2αu2 u∗ + αuxx = 0,
u∗t + 2αu(u∗ )2 − αu∗xx = 0,
where α was replaced by i after calculations. These three conservation laws are the
same as those given in [44].
64
Table 5.5: The first four conservation laws for the NLS equation (5.2).
Order
1
2
Density(ρ)
|u|2
uu∗xx − |u|4
Flux(J)
−i(uu∗x − u∗ ux )
−i(uu∗xxx − ux u∗xx − 4|u|2 uu∗x )
3
− i − u2 (u∗ )3 ux − 4u3 (u∗ )2 u∗x − u∗ u2x u∗x
2
3
1
+ uux (u∗x )2 − (u∗ )2 ux uxx + 2uu∗ u∗x uxx
2
4
9
1
3 ∗
∗
+ uu ux uxx + u2 u∗x u∗xx + u(u∗ )2 uxxx
2
2
4
1
2 ∗ ∗
∗
+ 2u u uxxx + ux uxxxx
4
1 ∗ − uu5x
4
3
2u3 (u∗ )3 − 6uu∗ ux u∗x
− 5u2 (u∗x )2 − u(u∗ )2 uxx
− 6u2 u∗ u∗xx + uu∗xxxx
i 24u3 (u∗ )4 ux + 6(u∗ )3 u3x + 64u4 (u∗ )3 u∗x
− 5u4 (u∗ )4 + 10u3 ×
× 5u∗ (u∗x )2 + 3(u∗ )2 u∗xx
+ u2 − 19(u∗xx )2
− 28u∗x u∗xxx + (u∗ )2 ux u∗x
4
− 10u∗ u∗xxxx + 8(u∗ )3 uxx
2
− 10|u| 2uxx u∗xx
+ u∗x uxxx
− u 19(u∗x )2 uxx − u∗6x
+ 20u∗ ux u∗xxx − 6(u∗ )3 u2x
+ (u∗ )2 uxxxx + 52ux u∗x u∗xx
− 18u(u∗ )2 u2x u∗x − 212u2 u∗ ux (u∗x )2
− 60u3 (u∗x )3 − 12u(u∗ )3 ux uxx
− 106u2 (u∗ )2 u∗x uxx − 19ux (u∗x )2 uxx
− 120u2 (u∗ )2 ux u∗xx − 216u3 u∗ u∗x u∗xx
− 52u2x u∗x u∗xx − 20u∗ ux uxx u∗xx
+ 110uu∗x uxx u∗xx + 52uux (u∗xx )2
− 10u2 (u∗ )3 uxxx − 10u∗ ux u∗x uxxx
+ 29u(u∗x )2 uxxx + 30uu∗ u∗xx uxxx
− 48u3 (u∗ )2 u∗xxx − 20u∗ u2x u∗xxx
+ 72uux u∗x u∗xxx + 40uu∗ uxx u∗xxx
+ 68u2 u∗xx u∗xxx − (u∗ )2 ux uxxxx
+ 12uu∗ u∗x uxxxx + 20uu∗ ux u∗xxxx
+ 40u2 u∗x u∗xxxx + u(u∗ )2 u5x + 12u2 u∗ u∗5x
∗
∗
+ ux u6x − uu7x
65
CHAPTER 6
CONCLUSIONS AND FURTHER RESEARCH
The preceding was an overview of construction methods for various properties
of integrable nonlinear PDEs. A novel method, based on scaling symmetry, was
discussed for the computation of Lax pairs, with good results. An old method for
constructing conservation laws was also discussed. Both methods used properties
inherent to the equations being discussed to turn a complicated problem into an algebraic problem. Both computational methods have been shown to work on a variety
of integrable nonlinear PDEs. Considering that most completely integrable PDEs are
closely-related to the ones discussed above, this is indicative of the usefulness of the
computational methods discussed.
This Thesis by no means closes the topics of research related to the construction
of Lax pairs and conservation laws as described above. Huge swaths of material were
ignored, or not discussed in this document. For example, this Thesis only focused on
2×2 matrix Lax pairs. However, n × n matrices are required for Lax pairs in higher
spatial dimensions. Moreover, both systems of equations and PDEs of dimension
higher than (1 + 1) were not adequately discussed. Both of these extensions would
provide ample research opportunity. There are obvious relationships between the
Drinfel’d-Sokolov method for conservation law computation and the method of Lax
pair computation presented in this Thesis which were not discussed. Either method
could be generalized by replacing current sums with Laurent-type expansions in λ
rather than Taylor-type expansions. Either method could be generalized by considering, more in-depth, non-polynomial PDEs. Furthermore, the problems associated
with weak and strong Lax pairs and their relationships with gauge transforms were
not adequately discussed. Finally, inverting the processes described above, to cre-
67
ate classes of PDEs related to a Lax pair or set of conservation laws, would be a
worthwhile undertaking.
68
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72
APPENDIX - OUTLINE OF DRINFEL’D-SOKOLOV ALGORITHM AND USE
OF MATHEMATICA
The goal of this program is to construct, simplify, and verify the validity of conservation laws for a given Lax Pair. To do this, we use the method proposed by Drinfel’d
and Sokolov as well as tools from calculus of variations described in Chapter 4. The
algorithm used to achieve this goal is as follows:
1. Construct conserved densities with Drinfel’d’s and Sokolov’s recurrence relationship, (4.6).
2. Verify that constructed densities are valid.
(a) Test if a density is trivial;
(b) Test that a density is exact.
3. Simplify densities
(a) Test for terms in density which are total divergences;
(b) Test for divergence-equivalence of linear combinations of terms.
4. Construct fluxes of conservation law based on densities.
Analytic methods of each of these steps are described above. Thus, all that
remains is to discuss the implementation of these methods in Mathematica.
1. Simplification Process
We use the following algorithm to remove divergence-equivalent terms from
the densities. In the description, let ρL be a list containing each term of the
conserved density ρ.
73
Step 1 Apply the Euler-operator to ρL . If any elements are identically zero in the
Euler space, remove the corresponding element from ρL .
Step 2 If there is only one element in ρL , do nothing to it and stop. The simplified
ρ is simply that element.
Step 3 Let S = {} and B = {} be empty sets. Let B contain the first non-zero
element of the list Lu ρL and let S contain the corresponding element of
ρL . This is the first element of the density.
Step 4 For an element ρi ∈ ρL , check if Lu ρi is in the span of B. If it is, let
B = B ∪ {Lu ρi } and S = S ∪ {ρi }. Otherwise, throw out element ρi .
Continue this process for all elements in ρL .
P
Step 5 Set the simplified ρ to ρsimp =
Si , where Si are the elements of S.
2. Coefficient Fixing
The goal of this process is to find coefficients for the elements of the density
which will make a density exact. We display some of the Mathematica code
used to do this before explaining it.
validDensitiesCoeff=Inner[Times,Table[c[i],{i,1,
Length[validDensities[i1]]}],
(validDensities[i1])/.Plus->List,Plus];
coeffList=stripper[EulerD[D[Expand[validDensitiesCoeff],t]
/.rule,u[x,t],{x,t}]];
solveRule=Flatten[Solve[Map[#==0&,Table[Coefficient
[EulerD[D[Expand[validDensitiesCoeff],t]/.rule,
u[x,t],{x,t}],coeffList[[i]]],{i,1,Length[coeffList]}]],
74
Table[c[i],{i,1,Length[validDensitiesCoeff]}]]];
validDensities[i1]=(validDensitiesCoeff/.solveRule)/.{c[i_]->1};
Step 1 First, put undetermined coefficients on each of the elements of the density, validDensities[i1]. This is done in two steps. First, the density
is converted to a list of elements. Then, the Inner function is used to
simultaneous multiply each element by an unknown coefficient and add
these elements together.
Step 2 Next, a stripper function is used to find which elements on the jet space
of u are present in the density.
Step 3 A list of equations to solve for each of the unknown coefficients is created.
This is done with the command Coefficient. This command finds the
coefficients of the density which are multiplying the same type of element
on the jet space of u.
Step 4 Finally, the list of equations created in Step 3 is solved and applied to the
density. Any still-unknown coefficients are normalized to one.
3. Normalization
The final piece of code involved the normalization of the highest-order terms
of a density. Again, we will first display the interesting bit of code, and then
explain it.
multiplier[i1,1]=(First[validDensities[i1,1]/.{Plus->List}])/
Cases[First[validDensities[i1,1]/.{Plus->List}],
Except[_Complex]][[1]];
Step 1 The density being considered is first converted to a list. The first element
of this list will always be the highest-order term due to Mathematica’s
75
ordering procedures, and so it is selected.
Step 2 The pattern matching function Cases is then used to find the elements
in this element which are not numbers. This is done with the Except
command.
Step 3 The highest order element is then divided by its non-numerical element to
find the coefficient which is in front of it.
The program used to construct these conservation laws is available from the author
upon request.
76