Abstract
The compatibility method is used for a generalized variable-coefficient Gardner equation (GVGE) with a forcing term. By the compatibility of the considered equation and a non-classical symmetry of a given form, four types of symmetry are obtained. Then, by solving the characteristic equations of symmetry, the GVGE is reduced to variable coefficients ordinary differential equations, and rich varieties of new similarity solutions are presented. Our results show that the compatibility method can be employed for variable coefficients nonlinear evolution equations with forcing terms.
Highlights
In the area of mathematics and physics, a considerable number of systems, ranging from gravitational dynamics and plasma dynamics to thermodynamics, can be modeled by nonlinear evolution equations
As shown in [27,28,29,30,31], the method is capable of obtaining abundant symmetry reductions and similarity solutions of the considered nonlinear evolution equations
Equation (1) becomes a constant-coefficient Gardner equation, which can be reduced to the following third-order ordinary differential equation: F + 2F2 F + FF + cF = 0, (38)
Summary
In the area of mathematics and physics, a considerable number of systems, ranging from gravitational dynamics and plasma dynamics to thermodynamics, can be modeled by nonlinear evolution equations. In 2006, a systematic method (named the compatibility method) was developed in [27] to seek non-classical symmetries and similarity solutions of a class of variable coefficients Zakharov–Kuznetsov equations. As shown in [27,28,29,30,31], the method is capable of obtaining abundant symmetry reductions and similarity solutions of the considered nonlinear evolution equations.
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