scholarly journals New similarity solutions for the generalized variable-coefficients KdV equation by using symmetry group method

2018 ◽  
Vol 25 (2) ◽  
pp. 66-70 ◽  
Author(s):  
Rehab M. El-Shiekh
Keyword(s):  
Symmetry Group ◽  
Kdv Equation ◽  
Variable Coefficients ◽  
Similarity Solutions ◽  
Group Method

EXTENDED SYMMETRIES AND SOLUTIONS OF (2 + 1)-DIMENSIONAL NONLINEAR SCHRÖDINGER EQUATION WITH VARIABLE COEFFICIENTS

2009 ◽  
Vol 20 (11) ◽  
pp. 1681-1696
Author(s):  
JIA WANG ◽  
BIAO LI
Keyword(s):  
Symmetry Group ◽  
Schrodinger Equation ◽  
Variable Coefficients ◽  
Group Method ◽  
Nonlinear Schrödinger ◽  
Time Space ◽  
Generalized Symmetry ◽  
Nonlinear Schrodinger Equations ◽  
Constant Coefficients ◽  
Dimensional Variable

By generalized symmetry group method, some time-space-dependent finite transformations between two different (2 + 1)-dimensional nonlinear Schrödinger equations (NLSE) are constructed. From these transformations, some (2 + 1)-dimensional variable coefficients NLSE can be reduced to another variable coefficients NLSE or corresponding constant coefficients NLSE. Abundant solutions of some (2 + 1)-dimensional variable coefficients NLSE are obtained from their corresponding constant coefficients NLSE.

Lie group analysis, Hamiltonian equations and conservation laws of Born–Infeld equation

2014 ◽  
Vol 07 (03) ◽  
pp. 1450040 ◽  
Author(s):  
Seyed Reza Hejazi
Keyword(s):  
Conservation Laws ◽  
Symmetry Group ◽  
Lie Group ◽  
Group Analysis ◽  
Similarity Solutions ◽  
Lie Symmetry ◽  
Group Method ◽  
Lie Group Analysis ◽  
Hamiltonian Equations

Lie symmetry group method is applied to study the Born–Infeld equation. The symmetry group is given, and similarity solutions associated to the symmetries are obtained. Finally the Hamiltonian equations including Hamiltonian symmetry group and conservation laws are determined.

Nonlocal Symmetries, Explicit Solutions, and Wave Structures for the Korteweg–de Vries Equation

2016 ◽  
Vol 71 (8) ◽  
pp. 735-740
Author(s):  
Zheng-Yi Ma ◽  
Jin-Xi Fei
Keyword(s):  
Vries Equation ◽  
Kdv Equation ◽  
Elliptic Functions ◽  
Group Method ◽  
Physical Interaction ◽  
Nonlocal Symmetries ◽  
Interaction Solutions ◽  
The Kdv Equation ◽  
De Vries ◽  
Exact Invariant

AbstractFrom the known Lax pair of the Korteweg–de Vries (KdV) equation, the Lie symmetry group method is successfully applied to find exact invariant solutions for the KdV equation with nonlocal symmetries by introducing two suitable auxiliary variables. Meanwhile, based on the prolonged system, the explicit analytic interaction solutions related to the hyperbolic and Jacobi elliptic functions are derived. Figures show the physical interaction between the cnoidal waves and a solitary wave.

Symmetry operators and exact solutions of a type of time-fractional Burgers–KdV equation

2019 ◽  
Vol 16 (02) ◽  
pp. 1950032 ◽  
Author(s):  
Azadeh Naderifard ◽  
S. Reza Hejazi ◽  
Elham Dastranj ◽  
Ahmad Motamednezhad
Keyword(s):  
Exact Solutions ◽  
Fourth Order ◽  
Vector Fields ◽  
Kdv Equation ◽  
Group Analysis ◽  
Invariant Subspaces ◽  
Similarity Solutions ◽  
Optimal System ◽  
Lie Point Symmetries ◽  
Symmetry Operators

In this paper, group analysis of the fourth-order time-fractional Burgers–Korteweg–de Vries (KdV) equation is considered. Geometric vector fields of Lie point symmetries of the equation are investigated and the corresponding optimal system is found. Similarity solutions of the equation are presented by using the obtained optimal system. Finally, a useful method called invariant subspaces is applied in order to find another solutions.

Similarity solutions of the super KdV equation

1995 ◽  
Vol 16 (9) ◽  
pp. 901-904 ◽  
Author(s):  
Yu Huidan ◽  
Zhang Jiefang
Keyword(s):  
Kdv Equation ◽  
Similarity Solutions ◽  
Super Kdv Equation

A Technique to Classify the Similarity Solutions of Nonlinear Partial (Integro-)Differential Equations. II. Full Optimal Subalgebraic Systems

1993 ◽  
Vol 48 (4) ◽  
pp. 535-550 ◽  
Author(s):  
H. Kötz
Keyword(s):  
Differential Equations ◽  
Symmetry Group ◽  
Lie Algebra ◽  
Three Dimensional ◽  
Classification Problem ◽  
Similarity Solutions ◽  
Point Symmetry ◽  
Point Symmetry Group ◽  
Relativistic Case ◽  
Lie Point Symmetry

"Optimal systems" of similarity solutions of a given system of nonlinear partial (integro-)differential equations which admits a finite-dimensional Lie point symmetry group Gare an effective systematic means to classify these group-invariant solutions since every other such solution can be derived from the members of the optimal systems. The classification problem for the similarity solutions leads to that of "constructing" optimal subalgebraic systems for the Lie algebra Gof the known symmetry group G. The methods for determining optimal systems of s-dimensional Lie subalgebras up to the dimension r of Gvary in case of 3 ≤ s ≤ r, depending on the solvability of G. If the r-dimensional Lie algebra Gof the infinitesimal symmetries is nonsolvable, in addition to the optimal subsystems of solvable subalgebras of Gone has to determine the optimal subsystems of semisimple subalgebras of Gin order to construct the full optimal systems of s-dimensional subalgebras of Gwith 3 ≤ s ≤ r. The techniques presented for this classification process are applied to the nonsolvable Lie algebra Gof the eight-dimensional Lie point symmetry group Gadmitted by the three-dimensional Vlasov-Maxwell equations for a multi-species plasma in the non-relativistic case.

scholarly journals Exact Solutions for a Third-Order KdV Equation with Variable Coefficients and Forcing Term

2009 ◽  
Vol 2009 ◽  
pp. 1-13 ◽  
Author(s):  
Alvaro H. Salas S ◽  
Cesar A. Gómez S
Keyword(s):  
Exact Solutions ◽  
Periodic Solutions ◽  
Riccati Equation ◽  
Function Method ◽  
Kdv Equation ◽  
Soliton Solutions ◽  
Variable Coefficients ◽  
Third Order ◽  
Forcing Term ◽  
Projective Riccati Equation Method

The general projective Riccati equation method and the Exp-function method are used to construct generalized soliton solutions and periodic solutions to special KdV equation with variable coefficients and forcing term.

Exact Soliton Solutions of the Variable Coefficient KdV Equation with Forced Term

2014 ◽  
Vol 548-549 ◽  
pp. 1196-1200
Author(s):  
Yong Mei Bao ◽  
Siqintana Bao
Keyword(s):  
Variable Coefficient ◽  
Evolution Equations ◽  
Nonlinear Evolution ◽  
Kdv Equation ◽  
Soliton Solutions ◽  
Nonlinear Ordinary Differential Equation ◽  
Variable Coefficients ◽  
Nonlinear Evolution Equations ◽  
Forced Term ◽  
Exact Soliton Solutions

In order to construct exact soliton solutions of nonlinear evolution equations with variable coefficients. By using a transformation, the variable coefficient KdV equation with forced Term is reduced to nonlinear ordinary differential equation (NLODE), after that, a number of exact solitons solutions of variable coefficient KdV equation with forced Term are obtained by using the equation shorted in NLODE. As it showed above, this kind of method can be applied in solving a large number of nonlinear evolution equations.

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