On the Redefinition of the Variational and `Partial' Variational Conservation Laws in a Class of Nonlinear PDEs with Mixed Derivatives

R Narain,A Kara

doi:10.3390/mca15040732

Abstract

The construction of conserved vectors using Noether’s and partial Noether’s theorems are carried out for high order PDEs with mixed derivatives. The resultant conserved flows display some interesting ‘divergence properties’ owing to the existence of the mixed derivatives. These are spelled out for various equations from mathematical physics.

Highlights

When considering the construction of conservation laws via Noether’s theorem using a Lagrangian or a ‘partial Lagrangian’, an interesting situation arises when the equations under investigation are such that the highest derivative term is mixed; the mixed derivative term is the one that involves differentiation by more than one of the independent variables
We consider the fourth-order Shallow Water Wave equations and the Camassa-Holms, Hunter-Saxton, Inviscid Burgers and KdV family of equations. These equations have their importance in many areas of physics, and real world applications, e.g., tsunamis which are characterized with long periods and wave lengths as a result they behave as shallow-water waves
L is referred to as a Lagrangian and a Noether symmetry operator X of L arises from a study of the invariance properties of the associated functional ∫

Summary

INTRODUCTION

When considering the construction of conservation laws via Noether’s theorem using a Lagrangian or a ‘partial Lagrangian’, an interesting situation arises when the equations under investigation are such that the highest derivative term is mixed; the mixed derivative term is the one that involves differentiation by more than one of the independent variables. When substituting the conserved flow back into the divergence relationship, a number of ‘extra’ terms (on which the Euler operator vanishes) arise. Consider an rth-order system of partial differential equations of n independent variables x = L is referred to as a Lagrangian and a Noether symmetry operator X of L arises from a study of the invariance properties of the associated functional ∫.

The divergence becomes

The total divergence is

3.DISCUSSION AND CONCLUSION