On symmetries and conservation laws of a Gardner equation involving arbitrary functions

In this paper, a simple way to construct exact solutions by using equivalence transformations is shown. We consider a generalized variable-coefficient Gardner equation from the point of view of Lie symmetries in partial differential equations. We obtain the continuous equivalence transformations of the equation in order to reduce the number of arbitrary functions and give a clearer formulation of the results. Furthermore, we calculate Lie symmetries of the reduced equation. Then, we determine the similarity variables and the similarity solutions which allow us to reduce our equation into an ordinary differential equation. Finally, we obtain some exact travelling wave solutions of the equation by using the simplest equation method.

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“…New symmetries were obtained for the case n ¤ 1 in [6] and, for n D 1 in [5], which are shown below:…”

Section: Classical Symmetries Of Class (6)mentioning

confidence: 99%

Travelling Wave Solutions of a Generalized Variable-Coefficient Gardner Equation

Rosa

Bruzón

2016

SEMA SIMAI Springer Series

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“…This method called the multiplier method allows finding all local conservation laws admitted by any evolution equation. Many papers have been published in the last few years using this method [11][12][13][14][15][16][17][18][19].…”

Section: Introductionmentioning

confidence: 99%

Symmetry Analysis and Conservation Laws of a Generalization of the Kelvin-Voigt Viscoelasticity Equation

Márquez

Bruzón

2019

Symmetry

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In this paper, we study a generalization of the well-known Kelvin-Voigt viscoelasticity equation describing the mechanical behaviour of viscoelasticity. We perform a Lie symmetry analysis. Hence, we obtain the Lie point symmetries of the equation, allowing us to transform the partial differential equation into an ordinary differential equation by using the symmetry reductions. Furthermore, we determine the conservation laws of this equation by applying the multiplier method.

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“…Specifically, the Gardner equation has recently been studied by different authors. [1][2][3][4][5][6][7] For instance, in Zhang et al, 7 a large number of solutions of a generalized Gardner equation were constructed. The authors obtained these solutions with the aid of 2 first-order nonlinear ordinary differential equations and a new generalized algebraic method.…”

Section: Introductionmentioning

confidence: 99%

“…A conservation law for class (1) is a space-time divergence expression that vanishes on all solutions of class (1). This concept has its origin in physics; nevertheless, it presents important applications in the study of differential equations or systems of differential equations.…”

Section: Introductionmentioning

confidence: 99%

On a generalized variable‐coefficient Gardner equation with linear damping and dissipative terms

Rosa

Recio

Garrido

et al. 2018

Math Methods in App Sciences

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Summary In this paper, a generalized variable‐coefficient Gardner equation which involves a linear damping term and a dissipative term is studied. This equation broadens out a large number of equations previously considered. The equivalence group is used to simplify the analysis of the equation. The gauging of arbitrary elements of the family allows us to perform an exhaustive study of the Lie symmetries admitted by the family. Finally, by means of the multiplier method, we classify all low‐order conservation laws of the equations that have been obtained by applying the gauging.

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On symmetries and conservation laws of a Gardner equation involving arbitrary functions

Cited by 9 publications

References 22 publications

Travelling Wave Solutions of a Generalized Variable-Coefficient Gardner Equation

Travelling Wave Solutions of a Generalized Variable-Coefficient Gardner Equation

Symmetry Analysis and Conservation Laws of a Generalization of the Kelvin-Voigt Viscoelasticity Equation

On a generalized variable‐coefficient Gardner equation with linear damping and dissipative terms

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