An Overview of the Generalized Gardner Equation: Symmetry Groups and Conservation Laws

  1. Bruzón, M. S. 1
  2. Gandarias, M. L. 1
  3. de la Rosa, R. 1
  1. 1 Departamento de Matemáticas, Universidad de Cádiz, Cádiz, Spain
Libro:
A Mathematical Modeling Approach from Nonlinear Dynamics to Complex Systems

Editorial: Springer

ISSN: 2195-9994,  2196-0003

ISBN: 9783319785110,  9783319785127

Año de publicación: 2018

Páginas: 7-26

Tipo: Capítulo de Libro

DOI: 10.1007/978-3-319-78512-7_2 GOOGLE SCHOLAR lock_openAcceso abierto editor

Resumen

In this paper we study the generalized variable-coefficient Gardner equations of the form u t  + A(t)f(u)u x  + C(t)f(u)2 u x  + B(t)u xxx  + Q(t)F(u) = 0. This family of equations includes many equations considered in the literature. Some conservation laws are derived by applying the multipliers method. The use of the equivalence group of this class allows us to perform an exhaustive study and a simple and clear formulation of the results. We study the equation from the point of view of Lie symmetries in partial differential equations. Finally, we calculate exact travelling wave solutions of the equation by using the simplest equation method.

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