Invariant Reduction for Partial Differential Equations. I: Conservation Laws and Systems with Two Independent Variables

TL;DR

This paper introduces an algorithmic reduction method for PDE systems with extended Kovalevskaya form, leveraging symmetries and conservation laws to find constants of motion for invariant solutions, with implementation in Maple.

Contribution

It presents a novel reduction procedure for PDEs that uses symmetries and conservation laws to compute invariants, demonstrated with examples and implemented in Maple.

Findings

The method effectively computes constants of motion for symmetry-invariant solutions.

Examples show the applicability to systems with point and higher symmetries.

Implementation in Maple facilitates practical use of the algorithm.

Abstract

For a system of partial differential equations that has an extended Kovalevskaya form, a reduction procedure is presented that allows one to use a local (point, contact, or higher) symmetry of a system and a symmetry-invariant conservation law to algorithmically calculate constants of motion holding for symmetry-invariant solutions. Several examples including cases of point and higher symmetry invariance are presented and discussed. An implementation of the algorithm in Maple is provided.