ABSTRACT (original article): Using a scaling symmetry, it is shown how to compute polynomial conservation laws, generalized symmetries, recursion operators, Lax pairs, and bilinear forms of polynomial nonlinear partial differential equations thereby establishing their complete integrability. The Gardner equation is chosen as the key example for it comprises both the Korteweg-de Vries and modified Korteweg de Vries (mKdV) equations. The Gardner and Miura transformations which connect these equations are also computed using the concept of scaling homogeneity. Exact solitary wave solutions and solitons of the Gardner equation are derived using Hirota’s method and other direct methods. The nature of these solutions depends on the sign of the cubic term in the Gardner equation and the underlying mKdV equation. It is shown that flat (table-top) waves of large amplitude only occur when the sign of the cubic nonlinearity is negative (defocusing case) whereas the focusing Gardner equation has the standard elastically colliding solitons. The paper’s aim is to provide a review of integrability properties and solutions of the Gardner equation and illustrate the applicability of the scaling symmetry approach. The methods and algorithms used in this paper have been implemented in Mathematica but can be adapted for major computer algebra systems. CITATION (original article): Willy Hereman, Ünal Göktaş, Using Symmetries to Investigate the Complete Integrability, Solitary Wave Solutions and Solitons of the Gardner Equation, arXiv:2408.07973, https://doi.org/10.48550/arXiv.2408.07973
Our paper is submitted to Mathematical and Computational Applications, Special Issue on Symmetry Methods for Solving Differential
Equations (34 pages, 9 figs., 75 refs.). The submission is also available on arXiv: https://arxiv.org/abs/2408.07973 and https://www.preprints.org/manuscript/202407.2297/v1 .
Equations (34 pages, 9 figs., 75 refs.). The submission is also available on arXiv: https://arxiv.org/abs/2408.07973 and https://www.preprints.org/manuscript/202407.2297/v1 .
Software used to generate the results in the paper are listed below with their corresponding public repositories where the codes and related documentation notebooks are available .
Painlevé Integrability Test
Painlevé Integrability Test
Our code and its documentation for performing the Painlevé integrability test is available at: https://github.com/willyhereman/PainleveTest-Mathematica/tree/main/V4
In[]:=
Get["PainleveTestV4-2018.m"]
Package PainleveTestV4.m was successfully loaded.
Last updated by Hereman on Saturday, June 30, 2018 (19:10).
Based on PainleveTestV2 from January 2009. Has updated pdeForm function.
Previously updated by Hereman on Sunday, January 25, 2009.
Previously updated by Baldwin on Friday, January 2, 2009.
Version 2 first released: February 1, 2006.
In[]:=
PainleveTestu[x,t]+αu[x,t]u[x,t]+βu[x,t]+u[x,t]0,u[x,t],{x,t},VerboseTrue
∂
t
∂
x
2
u[x,t]
∂
x
∂
{x,3}
The Painleve test is carried out for the following equation(s) in standard Mathematica notation:
(0,1) u (1,0) u 2 u[x,t] (1,0) u (3,0) u |
or, in more readable subscript notation, used to shorten the output:
u t u xxx u x u x 2 u[x,t] |
**************************************************************
Computation of THE DOMINANT BEHAVIORS.
The potential solutions {{alpha[1] -> -2}, {alpha[1] -> 0}} are being removed because there still is freedom, or they are greater than DominantBehaviorMax, or fail to balance highest exponent terms from (at least) two different terms in the given system. If > 0, then transformations like u -> 1/v, 1/(v^2), etc., could result in a system that might pass the Painleve test.
α
i
The dominant behaviors (including minimal degrees in g) are:
{{alpha[1]-1,alphaMin[1]-4}}
The system (within initialConstantsOfIntegration) (determining the first constant of integration) for the dominant behavior, {alpha[1] -> -1, alphaMin[1] -> -4} , is:
-β[x,t]-6u[1,0][x,t][x,t]0
3
u[1,0][x,t]
(1,0)
g
3
(1,0)
g
The solutions for u[i,0] are:
u[1,0][x,t]-,u[1,0][x,t]
[x,t]
6
(1,0)
g
β
[x,t]
6
(1,0)
g
β
FIRST SUMMARY: Dominant behaviors and first constants of integration (u[i,0]):
{alpha[1]-1,alphaMin[1]-4},u[1,0][x,t]-,{alpha[1]-1,alphaMin[1]-4},u[1,0][x,t]
6
g
x
β
6
g
x
β
*************************************************************
Computation of THE RESONANCES for VARIOUS DOMINANT BEHAVIORS
(one at the time).
**************************************************************
START OF A NEW BRANCH OF THE COMPUTATIONS.
The system (entering the function resonances`GenerateSystem) is:
++αu[x,t]+β0
u
t
u
xxx
u
x
u
x
2
u[x,t]
with dominant behavior (leading orders):
{alpha[1]-1,alphaMin[1]-4}
and corresponding coefficients:
u[1,0][x,t]-
[x,t]
6
(1,0)
g
β
The characteristic equation for the resonances is:
(-4+r)(-3+r)(1+r)β[x,t]0
3
(1,0)
g
The solutions (resonances r) of the characteristic equation are:
{r-1,r3,r4}
Computation of THE RESONANCES for VARIOUS DOMINANT BEHAVIORS
(one at the time).
**************************************************************
START OF A NEW BRANCH OF THE COMPUTATIONS.
The system (entering the function resonances`GenerateSystem) is:
++αu[x,t]+β0
u
t
u
xxx
u
x
u
x
2
u[x,t]
with dominant behavior (leading orders):
{alpha[1]-1,alphaMin[1]-4}
and corresponding coefficients:
u[1,0][x,t]
[x,t]
6
(1,0)
g
β
The characteristic equation for the resonances is:
(-4+r)(-3+r)(1+r)β[x,t]0
3
(1,0)
g
The solutions (resonances r) of the characteristic equation are:
{r-1,r3,r4}
SECOND SUMMARY: leading order behaviors and corresponding resonances:
{alpha[1]-1,alphaMin[1]-4},u[1,0][x,t]-,{r-1,r3,r4},{alpha[1]-1,alphaMin[1]-4},u[1,0][x,t],{r-1,r3,r4}
6
g
x
β
6
g
x
β
*************************************************************
Computation of THE CONSTANTS OF INTEGRATION and COMPATIBILITY CONDITIONS (if applicable) for the various dominant behaviors
(one at the time).
**************************************************************
START OF A NEW BRANCH OF THE COMPUTATIONS.
The system (entering the function constantsOfIntegration) is:
++αu[x,t]+β0
u
t
u
xxx
u
x
u
x
2
u[x,t]
Working with leading order behavior and resonances:
{alpha[1]-1,alphaMin[1]-4},u[1,0][x,t]-,{r-1,r3,r4}
[x,t]
6
(1,0)
g
β
Determining the integration constants AT LEVEL 1.
Solving the system (within constantsOfIntegration`Solve).
The solutions for are:
u
i,1
u[1,1][x,t]
α+
g
x
6
β
g
xx
2β
g
x
Working with leading order behavior and resonances:
{alpha[1]-1,alphaMin[1]-4},u[1,0][x,t]-,{r-1,r3,r4}
[x,t]
6
(1,0)
g
β
Determining the integration constants AT LEVEL 2.
Solving the system (within constantsOfIntegration`Solve).
The solutions for are:
u
i,2
u[1,2][x,t]-
4β--6β+4β
g
t
g
x
2
α
2
g
x
2
g
xx
g
x
g
xxx
4
6
3/2
β
3
g
x
Working with leading order behavior and resonances:
{alpha[1]-1,alphaMin[1]-4},u[1,0][x,t]-,{r-1,r3,r4}
[x,t]
6
(1,0)
g
β
Determining the integration constants AT LEVEL 3.
Solving the system (within constantsOfIntegration`NthResonance`Solve).
The solutions for are:
u
i,3
{u[1,3][x,t]u[1,3][x,t]}
Working with leading order behavior and resonances:
{alpha[1]-1,alphaMin[1]-4},u[1,0][x,t]-,{r-1,r3,r4}
[x,t]
6
(1,0)
g
β
Determining the integration constants AT LEVEL 4.
Solving the system (within constantsOfIntegration`NthResonance`Solve).
The solutions for are:
u
i,4
{u[1,4][x,t]u[1,4][x,t]}
Computation of THE CONSTANTS OF INTEGRATION and COMPATIBILITY CONDITIONS (if applicable) for the various dominant behaviors
(one at the time).
**************************************************************
START OF A NEW BRANCH OF THE COMPUTATIONS.
The system (entering the function constantsOfIntegration) is:
++αu[x,t]+β0
u
t
u
xxx
u
x
u
x
2
u[x,t]
Working with leading order behavior and resonances:
{alpha[1]-1,alphaMin[1]-4},u[1,0][x,t],{r-1,r3,r4}
[x,t]
6
(1,0)
g
β
Generalized Symmetries
Generalized Symmetries
Our Mathematica package InvariantsSymmetries.m for computation of invariants and symmetries of PDEs and DDEs
and its documentation is available on https://library.wolfram.com/infocenter/MathSource/570/.
and its documentation is available on https://library.wolfram.com/infocenter/MathSource/570/.
Special Solutions of PDEs
Special Solutions of PDEs
Our code for computation of special solutions of PDEs and its documentation is available at : https://github.com/willyhereman/PDESpecialSolutions-Mathematica/tree/main/V4. Note that our methods for Tanh and Sech type solutions are included in the Mathematica Kernel.
Recursion Operators of PDEs
Recursion Operators of PDEs
Our code for computation of recursion operators of PDEs and its documentation is available at: https://github.com/willyhereman/PDERecursionOperator-Mathematica
Out[]//InputForm=
{{{C[1]*operatorD[0] + operatorD[2] +
(2*α*operatorD[0]*u[x, t])/3 +
(2*β*operatorD[0]*u[x, t]^2)/3 +
(α*operatorD[-1]*Derivative[1, 0][u][x, t])/3 +
(2*β*operatorD[-1]*operatorD[0]*u[x, t]*
Derivative[1, 0][u][x, t])/3}}}
(2*α*operatorD[0]*u[x, t])/3 +
(2*β*operatorD[0]*u[x, t]^2)/3 +
(α*operatorD[-1]*Derivative[1, 0][u][x, t])/3 +
(2*β*operatorD[-1]*operatorD[0]*u[x, t]*
Derivative[1, 0][u][x, t])/3}}}
CITE THIS NOTEBOOK
CITE THIS NOTEBOOK
Using symmetries to investigate the complete integrability, solitary wave solutions and solitons of the Gardner equation
by Willy Hereman & Ünal Göktaş
Wolfram Community, STAFF PICKS, September 16, 2024
https://community.wolfram.com/groups/-/m/t/3275100
by Willy Hereman & Ünal Göktaş
Wolfram Community, STAFF PICKS, September 16, 2024
https://community.wolfram.com/groups/-/m/t/3275100