Moving boundary problems for a novel extended mKdV equation. Application of Ermakov-Painlev\'e II symmetry reduction

TL;DR

This paper introduces a new extended mKdV equation with symmetry reduction, providing exact solutions for moving boundary problems and exploring its embedding in broader nonlinear evolution equations with temporal modulation.

Contribution

It presents a novel extended mKdV equation with hybrid Ermakov-Painlevé II symmetry reduction and applies it to solve complex moving boundary problems exactly.

Findings

Exact Airy-type solutions for Stefan problems.

Embedding of extended mKdV in nonlinear evolution equations with temporal modulation.

Delimitation of temporal modulation in solitonic equations.

Abstract

A novel extension of the canonical solitonic mKdV equation is introduced which admits hybrid Ermakov-Painlev\'e II symmetry reduction. Application of the latter is made to obtain exact solution of Airy-type to a class of moving boundary problems of Stefan kind for this extended mKdV equation. A reciprocal transformation is then applied to the latter to generate an associated exactly solvable class of moving boundary problems for an extension of a base Casimir member of a compacton hierachy. The extended mKdV equation is shown to be embedded in a range of nonlinear evolution equations with temporal modulation as determined via the action of a class of involutory transformations with origin in Ermakov theory. Associated temporal modulation for the hybrid mKdV and KdV equation as embedded in the classical solitonic Gardner equation is delimited.