### Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 14 (2018), 017, 19 pages      arXiv:1712.01549      https://doi.org/10.3842/SIGMA.2018.017

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

Mikhail B. Sheftel a and Devrim Yazıcı b
a) Department of Physics, Boğaziçi University, Bebek, 34342 Istanbul, Turkey
b) Department of Physics, Yıldız Technical University, Esenler, 34220 Istanbul, Turkey

Received December 06, 2017, in final form March 02, 2018; Published online March 07, 2018

Abstract
We show that evolutionary Hirota type Euler-Lagrange equations in $(2+1)$ dimensions have a symplectic Monge-Ampère 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.

Key words: Lax pair; recursion operator; Hamiltonian operator; bi-Hamiltonian system.

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