Evolutionary Hirota Type (2+1)-Dimensional Equations: Lax Pairs, Recursion Operators and Bi-Hamiltonian Structures

TL;DR

This paper investigates (2+1)-dimensional evolutionary Hirota type equations, establishing their symplectic Monge-Ampère form, integrability via Lax pairs, and constructing Lagrangians, recursion operators, and multi-Hamiltonian structures.

Contribution

It introduces new integrable equations in (2+1) dimensions and develops their Lagrangian, recursion, and bi-Hamiltonian frameworks, expanding understanding of their geometric and algebraic properties.

Findings

Identification of symplectic Monge-Ampère form for these equations

Construction of Lax pairs for integrable cases

Development of bi-Hamiltonian and tri-Hamiltonian structures

Abstract

We show that evolutionary Hirota type Euler-Lagrange equations in (2+1) dimensions have a symplectic Monge-Amp\`ere form. We consider integrable equations of this type in the sense that they admit infinitely many hydrodynamic reductions and determine Lax pairs for them. For two seven-parameter families of integrable equations converted to two-component form we have constructed Lagrangians, recursion operators and bi-Hamiltonian representations. We have also presented a six-parameter family of tri-Hamiltonian systems.