Abstract not found

We present a project of classification of a certain class of bihamiltonian 1+1 PDEs depending on a small parameter. Our aim is to embed the theory of Gromov- Witten invariants of all genera into the theory of integrable systems. The project is focused at describing normal forms of the PDEs and their local bihamiltonian structures satisfying certain simple axioms. A Frobenius manifold or its degeneration is associated to every bihamiltonian structure of our type. The main result is a universal loop equation on the jet space of a semisimple Frobenius manifold that can be used for perturbative reconstruction of the integrable hierarchy. We show that first few terms of the perturbative expansion correctly reproduce the universal identities between intersection numbers of Gromov- Witten classes and their descendents.

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.

We develop a Hamiltonian theory for 2D soliton equations. In particular, we identify the spaces of doubly periodic operators on which a full hierarchy of commuting flows can be introduced, and show that these flows are Hamiltonian with respect to a universal symplectic form ω = 1 2 Res ∞ < Ψ ∗ 0 δL ∧ δΨ0> dk. We also construct other higher order symplectic forms and compare our formalism with the case of 1D solitons. Restricted to spaces of finite-gap solitons, the universal symplectic form agrees with the symplectic forms which have recently appeared in non-linear WKB theory, topological field theory, and Seiberg-Witten theories. We take the opportunity to survey some developments in these areas where symplectic forms have played a major role.

Geometrical optics limit of the Maxwell equations for nonlinear media with the Cole-Cole dependence of dielectric function and magnetic permeability on the frequency is considered. It is shown that for media with slow variation along one axis such a limit gives rise to the dispersionless Veselov-Novikov equation for the refractive index. It is demonstrated that the Veselov-Novikov hierarchy is amenable to the quasiclassical ¯ ∂-dressing method. Under more specific requirements for the media, one gets the dispersionless Kadomtsev-Petviashvili equation. Geometrical optics interpretation of some solutions of the above equations is discussed.

on the occasion of his 65th birthday. Abstract. We present a universal construction of almost duality for Frobenius manifolds. The analytic setup of this construction is described in details for the case of semisimple Frobenius manifolds. We illustrate the general considerations by examples from the singularity theory, mirror symmetry, the theory of Coxeter groups and Shephard groups, from the Seiberg- Witten duality. 1.

Integrability criterion for the Egorov hydrodynamic type systems is presented. The general solution by the generalized hodograph method is found. Examples are given. A description of three orthogonal curvilinear coordinate nets is discussed from the viewpoint of reciprocal transformations. In honour of Sergey Tsarev

Recently, a new model of propagation of the light through the so-called weakly three-dimensional Cole-Cole nonlinear medium with short-range nonlocality has been proposed. In particular, it has been shown that in the geometrical optics limit, the model is integrable and it is governed by the dispersionless Veselov-Novikov (dVN) equation. Burgers-Hopf equation can be obtained as 1+1-dimensional reduction of dVN equation. We discuss its properties in the specific context of nonlinear geometrical optics. An illustrative explicit example is considered. PACS numbers: 02.30.Ik, 42.15.Dp Key words: Nonlinear Optics, Integrable Systems. Many recent studies concerning the dispersionless integrable systems have shown their relevance in a broad variety of fields in physics such as Laplacian growth, topological field theory, nonlinear optics [1]-[10], as well as their deep

Centre de recherche mathématiques, Université de Montréal. We have generalized the approach in of Dunajski, Mason and Tod [10] and established a 1-1 correspondence between a solution of the universal Whitham hierarchy [23] and a twistor space. The twistor space consists of a complex surface and a family of complex curves together with a meromorphic 2-form. The solution of the Whitham hierarchy is given by deforming the curve in the surface. By treating the family of algebraic curves in CP 1 × CP 1 as a twistor space, we were able to express the deformations of the isomonodromic spectral curve in terms of the deformations generated by the Whitham hierarchy. 1

transformations