Bi-Hamiltonian representation, symmetries and integrals of mixed heavenly and Husain systems

TL;DR

This paper explores the bi-Hamiltonian structure, symmetries, and integrals of mixed heavenly and Husain systems, revealing their recursion operators, Lax pairs, and hierarchies of symmetries.

Contribution

It introduces new bi-Hamiltonian formulations, recursion operators, and symmetry hierarchies for the mixed heavenly and Husain equations, expanding understanding of their integrability.

Findings

Derived recursion operators and Lax pairs for the systems.

Established bi-Hamiltonian structures and symmetry hierarchies.

Identified all point and second-order symmetries and integrals.

Abstract

In the recent paper by one of the authors (MBS) and A. A. Malykh on the classification of second-order PDEs with four independent variables that possess partner symmetries (J. Phys. A: Math. Theor. Vol. 42 (2009) 395202 (20pp)), mixed heavenly equation and Husain equation appear as closely related canonical equations admitting partner symmetries. Here for the mixed heavenly equation and Husain equation, formulated in a two-component form, we present recursion operators, Lax pairs of Olver-Ibragimov-Shabat type and discover their Lagrangians, symplectic and bi-Hamiltonian structure. We obtain all point and second-order symmetries, integrals and bi-Hamiltonian representations of these systems and their symmetry flows together with infinite hierarchies of nonlocal higher symmetries.