Multi-Hamiltonian structure of Plebanski's second heavenly equation

Abstract

We show that Plebanski's second heavenly equation, when written as a first-order nonlinear evolutionary system, admits multi-Hamiltonian structure. Therefore by Magri's theorem it is a completely integrable system. Thus it is an example of a completely integrable system in four dimensions.

References in corpus (6)

• Hyper-K{ä}hler Hierarchies and their twistor theory
• Twistor theory of hyper-K{ä}hler metrics with hidden symmetries
• Partner symmetries and non-invariant solutions of four-dimensional heavenly equations
• Partner symmetries of the complex Monge-Ampere equation yield hyper-Kahler metrics without continuous symmetries
• Hamiltonian structure of real Monge-Ampère equations
• Covariant symplectic structure of the complex Monge-Ampère equation

Cited by in corpus (18)

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• Recursion operators for dispersionless integrable systems in any dimension
• Solvable vector nonlinear Riemann problems, exact implicit solutions of dispersionless PDEs and wave breaking
• A unified approach to computation of integrable structures
• Bi-Hamiltonian representation, symmetries and integrals of mixed heavenly and Husain systems
• Integrable dispersionless PDEs arising as commutation condition of pairs of vector fields
• Duality of positive and negative integrable hierarchies via relativistically invariant fields
• Recursion Operators and Tri-Hamiltonian Structure of the First Heavenly Equation of Plebański
• Self-dual gravity is completely integrable
• Recursion operators and bi-Hamiltonian structure of the general heavenly equation
• Evolutionary Hirota Type (2+1)-Dimensional Equations: Lax Pairs, Recursion Operators and Bi-Hamiltonian Structures
• Lax pairs, recursion operators and bi-Hamiltonian representations of (3+1)-dimensional Hirota type equations
• Integrability of dispersionless Hirota type equations in 4D and the symplectic Monge-Ampere property
• By Magri's Theorem, Self-Dual Gravity is Completely Integrable
• Recursion operators and bi-Hamiltonian representations of cubic evolutionary (2+1)-dimensional systems