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We consider the WDVV associativity equations in the four dimensional case. These nonlinear equations of third order can be written as a pair of six component commuting two-dimensional non-diagonalizable hydrodynamic type systems. We prove that these systems possess a compatible pair of local homogeneous Hamiltonian structures of Dubrovin--Novikov type (of first and third order, respectively).
The Oriented Associativity equation plays a fundamental role in the theory of Integrable Systems. In this paper we prove that the equation, besides being Hamiltonian with respect to a first-order Hamiltonian operator, has a third-order non-local homo
We study smooth SU(2) solutions of the Hitchin equations on R^2, with the determinant of the complex Higgs field being a polynomial of degree n. When n>=3, there are moduli spaces of solutions, in the sense that the natural L^2 metric is well-defined
Self-similar solutions of the so called Airy equations, equivalent to the dispersionless nonlinear Schrodinger equation written in Madelung coordinates, are found and studied from the point of view of complete integrability and of their role in the r
We introduce a bi-Hamiltonian hierarchy on the cotangent bundle of the real Lie group ${mathrm GL}(n,{mathbb{C}})$, and study its Poisson reduction with respect to the action of the product group ${{mathrm U}(n)} times {{mathrm U}(n)}$ arising from l
The Lagrangian representation of multi-Hamiltonian PDEs has been introduced by Y. Nutku and one of us (MVP). In this paper we focus on systems which are (at least) bi-Hamiltonian by a pair $A_1$, $A_2$, where $A_1$ is a hydrodynamic-type Hamiltonian