Non-Hamiltonian generalizations of the dispersionless 2DTL hierarchy

TL;DR

This paper introduces non-Hamiltonian integrable generalizations of the dispersionless 2DTL hierarchy, including new equations and a dressing scheme, expanding the understanding of integrable systems beyond Hamiltonian frameworks.

Contribution

It presents the first non-Hamiltonian two-component generalizations of the dispersionless 2DTL hierarchy, with new equations, transformations, and a dressing scheme.

Findings

Derived a simple two-component dispersionless 2DTL equation

Constructed a symmetric generalization of the elliptic 2DTL equation

Presented a dressing scheme based on the vector nonlinear Riemann problem

Abstract

We consider two-component integrable generalizations of the dispersionless 2DTL hierarchy connected with non-Hamiltonian vector fields, similar to the Manakov-Santini hierarchy generalizing the dKP hierarchy. They form a one-parametric family connected by hodograph type transformations. Generating equations and Lax-Sato equations are introduced, a dressing scheme based on the vector nonlinear Riemann problem is formulated. The simplest two-component generalization of the dispersionless 2DTL equation is derived, its differential reduction analogous to the Dunajski interpolating system is presented. A symmetric two-component generalization of the dispersionless elliptic 2DTL equation is also constructed.