Symmetry actions and brackets for adjoint-symmetries. II: Physical examples

TL;DR

This paper explores algebraic structures of symmetries and adjoint-symmetries in PDEs, introducing new actions and brackets, and illustrates these concepts through five complex physical PDE examples.

Contribution

It develops new algebraic frameworks for adjoint-symmetries and demonstrates their application to diverse nonlinear PDE systems in physics.

Findings

New symmetry actions on adjoint-symmetries established

A Lie bracket on adjoint-symmetries constructed

Applications to five physically relevant PDE systems

Abstract

Symmetries and adjoint-symmetries are two fundamental (coordinate-free) structures of PDE systems. Recent work has developed several new algebraic aspects of adjoint-symmetries: three fundamental actions of symmetries on adjoint-symmetries; a Lie bracket on the set of adjoint-symmetries given by the range of a symmetry action; a generalized Noether (pre-symplectic) operator constructed from any non-variational adjoint-symmetry. These results are illustrated here by considering five examples of physically interesting nonlinear PDE systems -- nonlinear reaction-diffusion equations, Navier-Stokes equations for compressible viscous fluid flow, surface-gravity water wave equations, coupled solitary wave equations, and a nonlinear acoustic equation.