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

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.

LaTeX2e source, 43 pages, 23 references, title modified, errors corrected, study of recursions of symmetries and integrals added

References in corpus (4)

• On the integrability of symplectic Monge-Ampére equations
• Lift of noninvariant solutions of heavenly equations from three to four dimensions and new ultra-hyperbolic metrics
• Integrable equations of the dispersionless Hirota type and hypersurfaces in the Lagrangian Grassmannian
• Lift of Invariant to Non-Invariant Solutions of Complex Monge-Ampère Equations

Cited by in corpus (6)

• Recursion operators for dispersionless integrable systems in any dimension
• Recursion Operators and Tri-Hamiltonian Structure of the First Heavenly Equation of Plebański
• Recursion operators and bi-Hamiltonian structure of the general heavenly equation
• Dispersionless Multi-Dimensional Integrable Systems and Related Conformal Structure Generating Equations of Mathematical Physics
• Integrability of dispersionless Hirota type equations in 4D and the symplectic Monge-Ampere property
• Lax pairs, recursion operators and bi-Hamiltonian representations of (3+1)-dimensional Hirota type equations