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Conservation laws for a generalized seventh order KdV equation

María de los Santos Bruzón, Almudena del Pilar Márquez, Tamara María Garrido, Elena Recio, Rafael de la Rosa

TL;DR

By applying the multiplier method, by applying the Lie method, a complete classification of low-order local conservation laws for a generalized seventh-order KdV equation depending on seven arbitrary nonzero parameters is obtained.

Abstract

In this paper, by applying the multiplier method we obtain a complete classification of low-order local conservation laws for a generalized seventh-order KdV equation depending on seven arbitrary nonzero parameters. We apply the Lie method in order to classify all point symmetries admitted by the equation in terms of the arbitrary parameters. We find that there are no special cases of the parameters for which the equation admits extra symmetries, other than those that can be found by inspection (scaling symmetry and space and time translation symmetries). We consider the reduced ordinary differential equations and we determined all integrating factors of the reduced equation from the combined $x$- and $t$- translation symmetries. Finally, we observe that all integrating factors arise by reduction of the low-order multipliers of the generalized seventh-order KdV equation.

Conservation laws for a generalized seventh order KdV equation

TL;DR

By applying the multiplier method, by applying the Lie method, a complete classification of low-order local conservation laws for a generalized seventh-order KdV equation depending on seven arbitrary nonzero parameters is obtained.

Abstract

In this paper, by applying the multiplier method we obtain a complete classification of low-order local conservation laws for a generalized seventh-order KdV equation depending on seven arbitrary nonzero parameters. We apply the Lie method in order to classify all point symmetries admitted by the equation in terms of the arbitrary parameters. We find that there are no special cases of the parameters for which the equation admits extra symmetries, other than those that can be found by inspection (scaling symmetry and space and time translation symmetries). We consider the reduced ordinary differential equations and we determined all integrating factors of the reduced equation from the combined - and - translation symmetries. Finally, we observe that all integrating factors arise by reduction of the low-order multipliers of the generalized seventh-order KdV equation.
Paper Structure (5 sections, 1 theorem, 19 equations)

This paper contains 5 sections, 1 theorem, 19 equations.

Table of Contents

  1. Introduction
  2. Multipliers and conservation laws of equation (\ref{['ed1']})
  3. Lie symmetries
  4. Integrating factors and first integrals for equation \ref{['ecf']}
  5. Conclusions

Key Result

Theorem 1

All nontrivial low-order local conservation laws admitted by the seventh-order KdV equation ed1 with $a_i\neq 0$, $i=1,...,7$ are given by:

Theorems & Definitions (1)

  • Theorem 1