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

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.