An Extended Modified Kadomtsov-Petviashvili Equation: Ermakov-Painlev\'e II Symmetry Reduction with Moving Boundary Application

TL;DR

This paper introduces a new 2+1-dimensional nonlinear evolution equation with temporal modulation that admits Ermakov-Painlevé II symmetry reduction and applies it to solve Stefan-type moving boundary problems.

Contribution

It extends involutory transformations to 2+1 dimensions and derives a broad class of equations with symmetry properties useful for moving boundary problems.

Findings

Derived exact solutions for Stefan-type moving boundary problems.

Extended symmetry reduction techniques to higher-dimensional nonlinear equations.

Demonstrated applicability to complex boundary problems with temporal modulation.

Abstract

Here, a novel 2+1-dimensional nonlinear evolution equation with temporal modulation is introduced which admits integrable Ermakov-Painlev\'e II symmetry reduction. Application is made to obtain exact solution to a class of Stefan-type moving boundary problems for this 2+1-dimensional nonlinear evolution equation. Involutory transformations with origin in autonomisation of certain Ermakov-type coupled systems are extended to 2+1-dimensions and applied to derive a wide 2+1-dimensional class with temporal modulation and which inherits the property of admittance of such hybrid Ermakov-Painlev\'e II symmetry reduction applicable to certain moving boundary problems.