Description of compatible differential-geometric Poisson brackets of the first order

Compatible local differential-geometric Poisson brackets of the first order (Dubrovin-Novikov type) are classified by solutions of modified Sinh-Gordon equation. It was proved by E.V. Ferapontov. In this paper integrable system describing compatible non-local differential-geometric Poisson brackets of the first order (Ferapontov type) is presented in case when one metric is flat and another one has co-dimension 1. This is a reduction of the Cherednik model (chiral fields).Comment: to appear in Theoretical and Mathematical Physic

The Boussinesq equation and Miura type transformations

A direct method for calculation of Miura type transformations via LA pair is used for the Boussinesq equation. Quadratic Miura type transformations connected with local weakly-nonlocal (Maltsev-Novikov) Hamiltonian structures. Modified systems are presented.Comment: to appear in Journal of Mathematical Science

Transformations of integrable hydrodynamic chains and their hydrodynamic reductions

Hydrodynamic reductions of the hydrodynamic chain associated with dispersionless limit of 2+1 Harry Dym equation are found by the Miura type and reciprocal transformations applied to the Benney hydrodynamic chain

Integrable Dispersive Chains and Energy Dependent Schrodinger Operator

In this paper we consider integrable dispersive chains associated with the so called Energy Dependent Schrodinger operator. In a general case multi component reductions of these dispersive chains are new integrable systems, which are characterised by two arbitrary natural numbers. Also we show that integrable three dimensional linearly degenerate quasilinear equations of a second order possess infinitely many differential constraints. Corresponding dispersive reductions are integrable systems associated with the Energy Dependent Schrodinger operator

Integrable hydrodynamic chains

A new approach for derivation of Benney-like momentum chains and integrable hydrodynamic type systems is presented. New integrable hydrodynamic chains are constructed, all their reductions are described and integrated. New (2+1) integrable hydrodynamic type systems are found.Comment: WARWICK CONFERENCE 2002 Geometry & Mechanics I

The Hamiltonian approach in classification and integrability of hydrodynamic chains

New approach in classification of integrable hydrodynamic chains is established. This is the method of the Hamiltonian hydrodynamic reductions. Simultaneously, this approach yields explicit Hamiltonian hydrodynamic reductions of the Hamiltonian hydrodynamic chains. The concept of reducible Poisson brackets is established. Also this approach is useful for non-Hamiltonian hydrodynamic chains. The deformed Benney hydrodynamic chain is considered

Explicit solutions of the WDVV equation determined by the "flat" hydrodynamic reductions of the Egorov hydrodynamic chains

Classification of the Egorov hydrodynamic chain and corresponding 2+1 quasilinear system is given in the previous paper. In this paper we present a general construction of explicit solutions for the WDVV equation associated with Hamiltonian hydrodynamic reductions of these Egorov hydrodynamic chain

The Kupershmidt hydrodynamic chains and lattices

This paper is devoted to the very important class of hydrodynamic chains first derived by B. Kupershmidt and later re-discovered by M. Blaszak. An infinite set of local Hamiltonian structures, hydrodynamic reductions parameterized by the hypergeometric function and reciprocal transformations for the Kupershmidt hydrodynamic chains are described

Modified dispersionless Veselov--Novikov equations and corresponding hydrodynamic chains

Various links connecting well-known hydrodynamic chains and corresponding 2+1 nonlinear equations are described

Integrability of the Egorov hydrodynamic type systems

Integrability criterion for the Egorov hydrodynamic type systems is presented. The general solution by generalized hodograph method is found. Examples are give