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Symmetries and conservation laws of a fifth-order KdV equation with time-dependent coefficients and linear damping

Rafael de la Rosa, María Luz Gandarias, María de los Santos Bruzón

Abstract

A fifth-order KdV equation with time dependent coefficients and linear damping has been studied. Symmetry groups have several different applications in the context of nonlinear differential equations. For instance, they can be used to determine conservation laws. We obtain the symmetries of the model applying Lie's classical method. The choice of some arbitrary functions of the equation by the equivalence transformation enhances the study of Lie symmetries of the equation. We have determined the subclasses of the equation which are nonlinearly self-adjoint. This allow us to obtain conservation laws by using a theorem proved by Ibragimov which is based on the concept of adjoint equation for nonlinear differential equations.

Symmetries and conservation laws of a fifth-order KdV equation with time-dependent coefficients and linear damping

Abstract

A fifth-order KdV equation with time dependent coefficients and linear damping has been studied. Symmetry groups have several different applications in the context of nonlinear differential equations. For instance, they can be used to determine conservation laws. We obtain the symmetries of the model applying Lie's classical method. The choice of some arbitrary functions of the equation by the equivalence transformation enhances the study of Lie symmetries of the equation. We have determined the subclasses of the equation which are nonlinearly self-adjoint. This allow us to obtain conservation laws by using a theorem proved by Ibragimov which is based on the concept of adjoint equation for nonlinear differential equations.
Paper Structure (12 sections, 3 theorems, 41 equations)

This paper contains 12 sections, 3 theorems, 41 equations.

Table of Contents

  1. Introduction
  2. Equivalence Transformations
  3. Lie classical symmetries of class (\ref{['eqTR']})
  4. Case 1: $F = 0$
  5. Case 2: $F = \hbox{constant} \neq 0$
  6. Formal Lagragian, adjoint equation and nonlinearly self-adjointness
  7. Conservation laws
  8. Case 1.
  9. Case 2.
  10. $F(t)=2$
  11. $F(t)=3$
  12. Conclusions

Key Result

Theorem 1

The adjoint equation to class (eqTR) is

Theorems & Definitions (5)

  • Definition 1
  • Theorem 1
  • Definition 2
  • Theorem 2
  • Theorem 3