On the hierarchy of partially invariant submodels of differential equations

TL;DR

This paper explores the hierarchical structure of partially invariant solutions (PISs) in differential equations, demonstrating how this hierarchy simplifies the classification process and providing a complete classification for ideal MHD equations.

Contribution

It introduces a hierarchical framework for PISs, proving the equivalence of two construction methods and applying it to classify solutions of ideal MHD equations.

Findings

Hierarchical structure reduces computational complexity in classifying PISs.

Two methods for constructing PISs are proven equivalent.

Complete classification of regular PISs for ideal MHD equations is provided.

Abstract

It is noticed, that partially invariant solution (PIS) of differential equations in many cases can be represented as an invariant reduction of some PIS of the higher rank. This introduce a hierarchic structure in the set of all PISs of a given system of differential equations. By using this structure one can significantly decrease an amount of calculations required in enumeration of all PISs for a given system of partially differential equations. An equivalence of the two-step and the direct ways of construction of PISs is proved. In this framework the complete classification of regular partially invariant solutions of ideal MHD equations is given.