On an integrable 2+1-dimensional extended Dym equation: Lax pair, $\bar{\partial}$-dressing scheme and modulation

TL;DR

This paper introduces a new 2+1-dimensional integrable extended Dym equation, constructs its Lax pair and dressing scheme, and develops modulated versions using classical involutory transformations.

Contribution

It presents the first 2+1-dimensional integrable extension of the Dym equation, along with its Lax pair, dressing scheme, and modulation methods.

Findings

Constructed Lax pair for the 2+1D extended Dym equation

Developed a $ar{ ext{d}}$-dressing scheme for solution generation

Generated integrable modulated equations using involutory transformations

Abstract

In 1+1-dimensions, an extension of the canonical solitonic Dym equation has previously been derived both in a geometric torsion evolution context and in the analysis of peakon solitonic phenomena in hydrodynamics. Here, a novel 2+1-dimensional S-integrable extended Dym-type equation is introduced. As Lax pair is constructed and an associated $\bar{\partial}$-dressing scheme detailed. Integrable modulated versions of the 2+1-dimensional extended Dym equation are generated via application of a class of involutory transformations with genesis in classical Ermakov theory.