Symbolic Computation of
Conservation Laws, Generalized Symmetries,
and Recursion Operators for
Nonlinear Differential-Difference Equations
Ünal Göktaş ∗ and Willy Hereman ◦
∗ Department of Computer Engineering
Turgut Özal University, Keçiören, Ankara, Turkey
unalgoktas@ttmail.com
◦ Department of Mathematical & Computer Sciences
Colorado School of Mines, Golden, Colorado, U.S.A.
whereman@mines.edu
3rd. Conference on Nonlinear Science and Complexity
Ankara, Turkey
Thursday, July 29, 2010, 17:00 p.m.
Acknowledgements
Mark Hickman (Univ. of Canterbury, New Zealand)
Bernard Deconinck (Univ. of Washington, U.S.A.)
Jan Sanders (Free University, Amsterdam,
The Netherlands)
Jing-Ping Wang (Univ. of Kent, Canterbury, U.K.)
Research supported in part by NSF
under Grant No. CCF-0830783
This presentation was made in TeXpower
Outline
•
Motivation
•
Definitions
I Systems of Differential-difference equations
I Dilation Invariance
I Up-shift and Down-shift Operators
I Conservation Law
I Generalized (Higher Order) Symmetry
I Recursion Operator
•
Algorithm for Conservation Laws
•
Algorithm for Generalized Symmetries
•
Algorithm for Recursion Operators
•
Computer Demonstration
Motivation
Differential-difference Equations (DDEs)
•
arise in key branches of physics (classical,
quantum, particle, statistical and plasma physics)
•
model wave phenomena in nonlinear optics
•
arise in bio-sciences
An integrable DDE
•
is linearizable or solvable via IST
•
has infinite sequence of conservation laws
•
has infinite sequence of generalized (higher order)
symmetries
•
has recursion operator(s) linking these symmetries
Systems of nonlinear DDEs
Consider a system of nonlinear DDEs of first order,
u̇n = F(un−` , ..., un−1 , un , un+1 , ..., un+m )
where un and F are vector-valued functions with N
components, and F is nonlinear function.
•
DDEs with one discrete variable, denoted by
integer n, which often corresponds to the
discretization of a space variable
•
dot stands for differentiation with respect to the
continuous variable (time t)
•
each component of F is assumed to be a
polynomial with constant coefficients
Leading Example: The Toda Lattice
One of the earliest and most famous examples of
completely integrable DDEs is the Toda lattice:
ÿn = exp (yn−1 − yn ) − exp (yn − yn+1 ),
where yn is the displacement from equilibrium of the
nth particle with unit mass under an exponential
decaying interaction force between nearest neighbors.
In new variables (un , vn ), defined by
un = ẏn , vn = exp (yn − yn+1 ), lattice can be written in
polynomial form
u̇n = vn−1 − vn ,
v̇n = vn (un − un+1 )
Dilation Invariance
A DDE is dilation invariant if it is invariant under a
dilation (scaling) symmetry.
Example
Toda Lattice is invariant under scaling symmetry
(t, un , vn ) → (λ−1 t, λ1 un , λ2 vn )
Uniformity in Rank
Define the weight, w, of a variable as the exponent of
the scaling parameter (λ) which multiplies that
variable. Since λ can be selected at will, t can always
d
be replaced by λt and, thus, w( dt
) = w(Dt ) = 1.
Weights of dependent variables are nonnegative,
rational, and independent of n. For example,
w(un−3 ) = · · · = w(un ) = · · · = w(un+2 ).
The rank, R, of a monomial is defined as the total
weight of the monomial. An expression is uniform in
rank if all of its terms have the same rank.
Dilation symmetries, which are special Lie-point
symmetries, are common to many DDEs.
Example
For the Toda lattice: w(un ) = 1, and w(vn ) = 2
Requiring uniformity in rank for each equation in the
Toda lattice allows one to compute the weights of the
dependent variables (and, thus, the scaling symmetry)
with simple linear algebra. Balancing the weights of
the various terms,
w(un ) + 1 = w(vn ),
w(vn ) + 1 = w(un ) + w(vn ),
yields
w(un ) = 1,
w(vn ) = 2
which confirms the dilation symmetry.
Up-Shift and Down-Shift Operator
Define the shift operator D by Dun = un+1 .
The operator D is often called the up-shift operator
or forward- or right-shift operator.
The inverse, D−1 , is the down-shift operator
or backward- or left-shift operator, D−1 un = un−1 .
Shift operators apply to functions by acting on the
arguments of the functions.
For example,
DF(un−` , · · · , un−1 , un , un+1 , · · · , un+m )
= F(Dun−` , · · · , Dun−1 , Dun , Dun+1 , . . . , Dun+m )
= F(un−`+1 , . . . , un , un+1 , un+2 , · · · , un+m+1 ).
Conservation Law
A conservation law of a DDE system
Dt ρ + ∆ J = 0
connects a conserved density ρ to an associated flux J,
where both are scalar functions depending on un and
its shifts.
•
Dt is the total derivative with respect to time,
•
∆ = D − I is the forward difference operator,
•
I is the identity operator
For readability, the components of un will be denoted
by un , vn , wn , etc.
Example
The Toda lattice has infinitely many conservation laws
List of the densities of rank 1 ≤ R ≤ 4 :
ρ(1) = un
ρ(2) =
ρ(3) =
ρ(4) =
1 2
u + vn
2 n
1 3
u + un (vn−1 +
3 n
2
1 4
u
+
u
n (vn−1 +
4 n
+ 21 vn2 + vn vn+1
vn )
vn ) + un un+1 vn
The first two density-flux pairs are easily computed by
hand, and so is
ρ(0)
n = ln(vn )
which is the only non-polynomial density (of rank 0).
Generalized Symmetry
A vector function G(un ) is called a generalized
symmetry of DDE system if the infinitesimal
transformation un → un + G leaves the DDE system
invariant up to order . G must then satisfy
Dt G = F0 (un )[G]
on solutions of the DDE system, where F0 (un )[G] is the
Fréchet derivative of F in the direction of G.
In the vector case with, say, components un and vn ,
the Fréchet derivative operator is a matrix operator:


P ∂F1 k P ∂F1 k

k ∂un+k D
k ∂vn+k D 


0
.
F (un ) = 


 P ∂F

P
∂F
k
k
2
2
D
D
k ∂u
k ∂v
n+k
n+k
Applied to G = (G1 G2 )T , where T is transpose, one
obtains
X ∂Fi
X ∂Fi
Fi 0 (un )[G] =
Dk G1 +
D k G2 ,
∂un+k
∂vn+k
k
k
with i = 1, 2.
Example
The first two non-trivial symmetries of the Toda
lattice are


vn − vn−1
(1)
,
G
= 
vn (un+1 − un )


vn (un + un+1 ) − vn−1 (un−1 + un )
(2)
.
G
= 
vn (u2n+1 − u2n + vn+1 − vn−1 )
Recursion Operator
A recursion operator R connects symmetries
G(j+s) = RG(j) ,
where j = 1, 2, · · · , and s is the gap length.
The symmetries are linked consecutively if s = 1.
This happens in most (but not all) cases. For
N -component systems, R is an N × N matrix operator.
The defining equation for R is
Dt R + [R, F0 (un )]
∂R
=
+ R0 [F] + R ◦ F0 (un ) − F0 (un ) ◦ R = 0,
∂t
where [ , ] denotes the commutator and ◦ the
composition of operators.
R0 [F] is the Fréchet derivative of R in the direction of
F. For the scalar case, the operator R is of the form
R = U (un ) O((D − I)−1 , D−1 , I, D) V (un ),
and then
X
X
∂U
∂V
0
k
k
R [F ] =
(D F )
OV +
U O(D F )
.
∂un+k
∂un+k
k
k
For the vector case, the elements of the N × N
operator matrix R are of the form
Rij = Uij (un ) Oij ((D − I)−1 , D−1 , I, D) Vij (un ).
Hence, for the 2-component case
0
R [F]ij
∂Uij
Oij Vij
=
(D F1 )
∂un+k
k
X
∂Uij
k
+
(D F2 )
Oij Vij
∂vn+k
k
X
∂Vij
k
+
Uij Oij (D F1 )
∂u
n+k
k
X
∂Vij
k
+
Uij Oij (D F2 )
.
∂vn+k
k
X
k
Example
The recursion operator of the Toda lattice is


un I
D−1 +I+(vn −vn−1 )(D−I)−1 v1n I




R =
.


vn I+vn D un+1 I+vn (un+1 −un )(D−I)−1 v1n I
It is straightforward to verify that RG(1) = G(2) .
Algorithm for Conservation Laws
As an example, we will compute the density ρ(3)
(of rank R = 3).
Step 1: Construct the Form of the Density
Start from V = {un , vn }, the set of dependent variables
with weights.
List all monomials in u and v of rank R = 3 or less:
M = {u3n , u2n , un vn , un , vn }. Next, for each monomial in
M, introduce the correct number of t-derivatives so
that each term has rank 3.
Using the Toda lattice, compute
0u v
d0 u3n
d
n n
3
=
u
,
=
u
v
,
n
n
n
dt0
dt0
du2n
= 2un u̇n = 2un vn−1 − 2un vn ,
dt
dvn
= v̇n = un vn − un+1 vn ,
dt
d2 un
du̇n
d(vn−1 − vn )
=
=
2
dt
dt
dt
= un−1 vn−1 − un vn−1 − un vn + un+1 vn .
Gather the terms in the right hand sides to get
R = {u3n , un vn−1 , un vn , un−1 vn−1 , un+1 vn }.
Identify members belonging to the same equivalence
classes and replace them by their canonical
representatives. For example, un vn−1 ≡ un+1 vn .
Adhering to lexicographical ordering, use un vn−1
instead of un+1 vn .
Doing so, replace R by S = {u3n , un vn−1 , un vn }, which
has the building blocks of the density. Linearly
combine the monomials in S with undetermined
coefficients ci to get the candidate density of rank 3 :
ρ = c1 u3n + c2 un vn−1 + c3 un vn .
Step 2: Compute the Undetermined Coefficients ci
Compute Dt ρ and use the Toda lattice to eliminate u̇n
and v̇n and their shifts. Next, introduce the main
representatives to get
E = (3c1 −c2 )u2n vn−1 +(c3 −3c1 )u2n vn +(c3 −c2 )vn vn+1
+(c2 − c3 )un un+1 vn +(c2 − c3 )vn2 + ∆J,
with
2
J = (c3 − c2 )vn−1 vn + c2 un−1 un vn−1 + c2 vn−1
.
Set E − ∆J ≡ 0 to get the linear system
3c1 − c2 = 0,
c3 − 3c1 = 0,
c2 − c3 = 0.
Select c1 = 13 and substitute the solution
c1 = 13 , c2 = c3 = 1, into the candidate form to obtain
density
ρ(3) = 13 u3n + un vn−1 + un vn ,
with matching flux
2
J (3) = un−1 un vn−1 + vn−1
.
Algorithm for Symmetries
As an example, we will compute the symmetry
(2)
(2)
(2)
G = (G1 , G2 ) of rank (3, 4).
Step 1: Construct the Form of the Symmetry
Listing all monomials in un and vn of ranks 3 and 4,
or less:
L1 = {u3n , u2n , un vn , un , vn },
L2 = {u4n , u3n , u2n vn , u2n , un vn , un , vn2 , vn }.
Next, for each monomial in L1 and L2 , introduce the
necessary t-derivatives so that each term exactly has
ranks 3 and 4, respectively. At the same time, use the
Toda lattice to remove all t−derivatives.
Doing so, based on L1 ,
0
d0 3
d
3
(u
)
=
u
n
n,
0
0 (un vn ) = un vn ,
dt
dt
d 2
(un ) = 2un u̇n = 2un vn−1 − 2un vn ,
dt
d
(vn ) = v̇n = un vn − un+1 vn ,
dt
d2
d
d
(u̇n ) =
(vn−1 − vn )
2 (un ) =
dt
dt
dt
= un−1 vn−1 − un vn−1 − un vn + un+1 vn .
Put the terms from the right hand sides into a set:
R1 = {u3n , un−1 vn−1 , un vn−1 , un vn , un+1 vn }.
Similarly, based on the monomials in L2 , construct
2
,
R2 = {u4n , u2n−1 vn−1 , un−1 un vn−1 , u2n vn−1 , vn−2 vn−1 , vn−1
u2n vn , un un+1 vn , u2n+1 vn , vn−1 vn , vn2 , vn vn+1 }.
Linearly combine the monomials in R1 and R2 with
undetermined coefficients ci to get the form of the
components of the candidate symmetry:
(2)
G1
= c1 u3n + c2 un−1 vn−1 + c3 un vn−1 + c4 un vn
+c5 un+1 vn ,
(2)
G2
= c6 u4n + c7 u2n−1 vn−1 + c8 un−1 un vn−1 + c9 u2n vn−1
2
+c10 vn−2 vn−1 + c11 vn−1
+ c12 u2n vn + c13 un un+1 vn
+c14 u2n+1 vn + c15 vn−1 vn + c16 vn2 + c17 vn vn+1 .
Step 2: Compute the Undetermined Coefficients ci
To determine the coefficients ci , require that the
symmetry condition holds on any solution of the DDE.
Compute Dt G and use the DDE to remove all
u̇n−1 , u̇n , u̇n+1 , etc.
Compute the Fréchet derivative and, in view of the
symmetry condition, equate the resulting expressions.
Treat as independent all the monomials in un and their
shifts, to obtain the linear system that determines the
coefficients ci .
Apply the strategy to the Toda lattice with candidate
form of the symmetry, to get
c1 = c6 = c7 = c8 = c9 = c10 = c11 = c13 = c16 = 0,
−c2 = −c3 = c4 = c5 = −c12 = c14 = −c15 = c17 .
Set c17 = 1 and substitute into the candidate form to
(2)
(2)
(2)
get G = (G1 , G2 ), as given earlier.
Algorithm for Recursion Operators
We will now construct the recursion operator the
recursion operator for the Toda lattice.
Step 1: Determine the Rank of the Recursion Operator
The difference in the ranks of symmetries is used to
compute the rank of the elements of the recursion
operator. Recall that
 
 
2
3
(1)
(2)



.
rank G =
, rank G =
3
4
Assume that R G(1) = G(2) and use the formula
(k+1)
rank Rij = rank Gi
(k)
− rank Gj ,
to compute a rank matrix associated to operator R :


1 0
.
rank R = 
2 1
Step 2: Determine the Form of the Recursion Operator
Assume that R = R0 + R1 . R0 is a sum of terms
involving D−1 , I, and D. The coefficients of these terms
are admissible power combinations of un , un+1 , vn , and
vn−1 (which come from the terms on the right hand
sides of the Toda lattice), so that all the terms have
the correct rank. The maximum up-shift and
down-shift operator that should be included can be
determined by comparing two consecutive symmetries.
For the Toda lattice,


(R0 )11 (R0 )12

,
R0 =
(R0 )21 (R0 )22
with
(R0 )11 = (c1 un + c2 un+1 ) I,
(R0 )12 = c3 D−1 + c4 I,
(R0 )21 = (c5 u2n + c6 un un+1 + c7 u2n+1 + c8 vn−1 + c9 vn ) I
+(c10 u2n + c11 un un+1 + c12 u2n+1 + c13 vn−1
+c14 vn ) D,
(R0 )22 = (c15 un + c16 un+1 ) I.
R1 is a linear combination (with undetermined
coefficients c̃jk ) of all suitable products of symmetries
and covariants, i.e. Fréchet derivatives of densities,
sandwiching (D − I)−1 . Hence,
XX
(j)
−1
(k)0
c̃jk G (D − I) ⊗ ρn ,
j
k
where ⊗ denotes the matrix outer product, defined as


(j)
G1
(k)0

 (D−I)−1 ⊗ ρ(k)0
ρ
=
n,1
n,2
(j)
G2


(j)
(k)0
(j)
(k)0
G1 (D−I)−1 ρn,1 G1 (D−I)−1 ρn,2

.
(j)
(k)0
(j)
(k)0
G2 (D−I)−1 ρn,1 G2 (D−I)−1 ρn,2
(0)0
(1)
(G , ρn )
Only the pair
can be used, otherwise the
ranks in the recursion operator would be exceeded.
Computing
1
ρ(0)0
=
,
0
I
n
vn
After renaming c̃10 to c17 , obtain


0
c17 (vn−1 −vn )(D−I)−1 v1n I
.
R1 = 
0 c17 vn (un −un+1 )(D−I)−1 v1n I
Then,
R = R0 + R1 .
Step 3: Determine the unknown coefficients
Compute all the terms in the defining equation to find
the ci . We refer to our papers for the details of the
computation, resulting in
c2 = c5 = c6 = c7 = c8 = c10 = c11 = c12 = c13 = c15 =
0, c1 = c3 = c4 = c9 = c14 = c16 = 1, and c17 = −1.
Substitute the constants into the candidate recursion
operator to get its final form:


un I
D−1 +I+(vn −vn−1 )(D−I)−1 v1n I




R =
.


vn I+vn D un+1 I+vn (un+1 −un )(D−I)−1 v1n I
Software Demonstration
Software packages in Mathematica
Codes are available via the Internet:
URL: http://inside.mines.edu/∼whereman/
Thank You