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Equivalence transformations and conservation laws for a generalized variable-coefficient Gardner equation

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

TL;DR

This class broadens out many other equations previously considered and construction of conservation laws are derived for the nonlinearly self-adjoint equations by using a general theorem on conservation laws.

Abstract

In this paper we study the generalized variable-coefficient Gardner equations of the form $u_t + A(t)u^n\,u_x+ C(t)\,u^{2n}u_x + B(t)\,u_{xxx} + Q(t)\,u =0$. This class broadens out many other equations previously considered: Johnpillai and Khalique (2010), Molati and Ramollo (2012) and Vaneeva, Kuriksha and Sophocleous (2015). Equivalence group of the class under consideration has been constructed which permit an exhaustive study and a simple and clear formulation of the results. Some conservation laws are derived for the nonlinearly self-adjoint equations, based on differential substitutions, and by using the direct method of the multipliers.

Equivalence transformations and conservation laws for a generalized variable-coefficient Gardner equation

TL;DR

This class broadens out many other equations previously considered and construction of conservation laws are derived for the nonlinearly self-adjoint equations by using a general theorem on conservation laws.

Abstract

In this paper we study the generalized variable-coefficient Gardner equations of the form . This class broadens out many other equations previously considered: Johnpillai and Khalique (2010), Molati and Ramollo (2012) and Vaneeva, Kuriksha and Sophocleous (2015). Equivalence group of the class under consideration has been constructed which permit an exhaustive study and a simple and clear formulation of the results. Some conservation laws are derived for the nonlinearly self-adjoint equations, based on differential substitutions, and by using the direct method of the multipliers.
Paper Structure (12 sections, 4 theorems, 70 equations)

This paper contains 12 sections, 4 theorems, 70 equations.

Table of Contents

  1. Introduction
  2. Equivalence transformations
  3. Classical Symmetries of class (\ref{['eqtrans']})
  4. Case 1: $n \neq 1$, $A(t)= a \,(b \, t +c)^{-\frac{1}{3}}$, $Q(t)=d \, (b \, t+c)^{-1}$
  5. Case 2: $n=1$, $A(t)= a \,(b \, t +c)^{-d}$, $Q(t)=b \,d \, (b \, t+c)^{-1}$
  6. Case 3: $n={\frac{1}{2}}$, $A(t)=0$, $Q(t) \hbox{arbitrary}$
  7. Formal Lagrangian and adjoint equation
  8. Nonlinearly self-adjoint equations
  9. Conservation Laws
  10. Conservation laws by using the general theorem by Ibragimov
  11. Conservation laws by using the direct method of the multipliers of Bluman and Anco
  12. Conclusions

Key Result

Theorem 1

The equivalence group of class (ed1) consists of the transformations where $\epsilon_1$, $\epsilon_2$ and $\epsilon_r$ are arbitrary constants.

Theorems & Definitions (5)

  • Theorem 1
  • Theorem 2
  • Definition 1
  • Theorem 3
  • Theorem 4