On mKdV and associated classes of moving boundary problems: reciprocal connections

Abstract

A class of Stefan-type moving boundary problems for the canonical modified Korteweg–de Vries (mKdV) equation of soliton theory is solved via application of a similarity reduction to Painlevé II which involves Airy’s equation. A reciprocal transformation is applied to derive a linked class of solvable moving boundary problems for a basic Casimir member of a compacton hierarchy. Application of a class of involutory transformations with origin in an autonomisation procedure for the Ermakov–Ray–Reid system is then used to isolate novel solvable moving boundary problems for Ermakov-modulated mkdV equations.