ASymbolicAlgorithmfortheComputationofConservationLaws
fromLaxPairs
Jacob Rezac
Willy Hereman
University of Delaware
Colorado School of Mines
NSF research award no. CCF-0830783
Motivation
Conservation Law Construction
We demonstrate a systematic method for the calculation of innitelymany conservation laws of a completely integrable partial dierential
Manipulate
Example
Key Idea
X and T so that [X, T ] = 0.
The Korteweg-de Vries equation,
ut + 6uux + uxxx = 0,
equation (PDE). We're interested because conservation laws:
•
•
•
suggest the complete integrability of a PDE;
for
Assume a PDE has the Lax pair
X = X0 + λX1
X0
with constant entries.
o-diagonal and diagonal
X1
and
T =
Pm
i
T
λ
i=0 i
has a Lax pair [3] with
are useful for numerical and physical applications;
Step 1: Let
P0 = X0
and recursively solve
imply a Lax pair is strong [1].
Pn + [Γn+1 , X1 ] = Γn X0 −
0
Γn
−
n−1
X
X=
Pk Γn−k
Hard to calculate;
for the unknowns
Unwieldy (see gure).
Pn
and
Γn .
The diagonal entries of of
We extend a result from Drinfel'd and Sokolov in [2] to calculate
Pn
are conserved densities,
ρn .
Step 2: Compute
Jn =
Preliminaries
to nd the uxes
Consider the PDE
1
P1 = i
2
0
−1
u
0
−u 0
0 u
1
P3 = i
8
,
T
on the jet space of
(ρn )t dx
are said to be a
Jn .
Z
J1 =
2
.
u + uxx
0
0
−u2
.
Z
(ρ1 )t dx = −
2
6uux + uxx dx = −3u − uxx
and, similarly,
J2 = −4u +
150
2
ux
− 2uuxx .
Similar results are seen, for example, with the modied Korteweg-de
Lax pair if
Vries equation, the sin- and sinh-Gordon equations, and the nonlinear
Xt − Tx + [X, T ] = 0
Schrodinger equation. Further work is needed to adjust the algorithm
on (1).
If (1) is such that functions
ρ(x, t)
and
J(x, t)
to more general PDEs.
100
satisfy
References
ρt − Jx = 0,
then the PDE is said to admit a conservation law with density
ux
0
−i
The uxes are
(1)
u(x, t)
i
0
ρ3 = u2
3
and
−λ
ρ1 = u
Step 3: Reduce to simplest form.
ut = f (u, ux , uxx , . . .).
X
and
Cleaning up,
Z
conservation laws given a Lax pair.
Matrix functions
Ti λi
The recurrence relationship gives, e.g.,
k=0
•
T =
of the form
i=0
Problems with Conservation Laws:
•
T
3
P
ρ
and
[1] S.Y. Sakovich. On zero-curvature representations of evolution equations.
50
Journal of Physics A: Mathematical and General, 28(10):2861-2869, 1995.
J.
Lax pairs are gauge invariant. Given a nonsingular matrix
X̃ = GXG
−1
−1
+ Gx G
and
are Lax pairs for the same PDE as
X
−1
T̃ = GT G
and
T.
[2] V.G. Drinfel'd and V.V. Sokolov. Lie algebras and equations of Korteweg-
G,
de Vries type. Journal of Mathematical Sciences, 30(2):1975-2036, 1985.
−1
+ Gt G
2
Number of terms in
4
nth
6
8
10
conservation law for the Korteweg-de Vries
equation (blue) and number of seconds to simultaneously compute
these rst
n
conservation laws using unoptimized Mathematica code
(red).
Translated from original Russian.
[3] M.J. Ablowitz and H. Segur.
form.
Solitons and the Inverse Scattering Trans-
SIAM, Philadelphia, PA, 1981.