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

TL;DR

The paper introduces a novel 2+1-dimensional extension of the Dym equation, $u_t+2\partial_x(1-\partial_{xx})\left(1/u^{1/2}\right)+6u^2\left[u^{-1}\partial_x^{-1}(u^{1/2})_y\right]_y=0$, and establishes its integrable structure via a Lax pair and a $\bar{\partial}$-dressing scheme. It analyzes linear representations, derives a potential formulation with $W_x=1/V$, $W_y=\rho$, and shows how a pair of equations for a potential $f$ (equations (45)-(46)) encode the extended Dym dynamics, with $f=W$ linking to the potential equation. Modulation of the system is developed through a suite of involutory transformations, including spatial and temporal modulations, as well as Ermakov and Ermakov-Painlevé schemes, producing a broad class of S-integrable, modulated 2+1D Dym equations. The framework unifies reciprocal, gauge, and Ermakov-type symmetries to generate analytically tractable modulated models, with potential implications for nonlinear wave dynamics and related physical contexts. Overall, the work extends the Dym family to 2+1 dimensions and provides a versatile modulation toolkit grounded in $\bar{\partial}$-dressing and Lax pair formalisms.

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.