Recursion operators and bi-Hamiltonian representations of cubic evolutionary (2+1)-dimensional systems

TL;DR

This paper constructs and analyzes (2+1)-dimensional PDEs with second-order derivatives, revealing new bi-Hamiltonian systems and providing Lax pairs, recursion operators, and Hamiltonian structures for integrable cubic equations.

Contribution

It introduces a method to find bi-Hamiltonian structures and recursion operators for a class of cubic (2+1)-dimensional PDEs, including three new integrable systems.

Findings

Derived all (2+1)-dimensional PDEs with Euler-Lagrange form depending on second derivatives.

Established Hamiltonian and Lax pair representations for these systems.

Discovered three new bi-Hamiltonian (2+1)-dimensional systems.

Abstract

We construct all (2+1)-dimensional PDEs depending only on 2nd-order derivatives of unknown which have the Euler-Lagrange form and determine the corresponding Lagrangians. We convert these equations and their Lagrangians to two-component forms and find Hamiltonian representations of all these systems using Dirac's theory of constraints. We consider three-parameter integrable equations that are cubic in partial derivatives of the unknown applying our method of skew factorization of the symmetry condition. Lax pairs and recursion relations for symmetries are determined both for one-component and two-component forms. For cubic three-parameter equations in the two-component form we obtain recursion operators in $2\times 2$ matrix form and bi-Hamiltonian representations, thus discovering three new bi-Hamiltonian (2+1) systems.