Huygens' Principle in Minkowski Spaces and Soliton Solutions of the Korteweg-de Vries Equation

A new class of linear second order hyperbolic partial differential operators satisfying Huygens' principle in Minkowski spaces is presented. The construction reveals a direct connection between Huygens' principle and the theory of solitary wave solutions of the Korteweg-de Vries equation.Comment: 23 pages, LaTeX, to be published in Comm.Math.Phys (1997

On Integrability and Exact Solvability in Deterministic and Stochastic Laplacian Growth

We review applications of theory of classical and quantum integrable systems to the free-boundary problems of fluid mechanics as well as to corresponding problems of statistical mechanics. We also review important exact results obtained in the theory of multi-fractal spectra of the stochastic models related to the Laplacian growth: Schramm-Loewner and Levy-Loewner evolutions

On Bose-Einstein condensate inside moving exciton-phonon droplets

We explore a nonlinear field model to describe the interplay between the ability of excitons to be Bose condensed and their interaction with other modes of a crystal. We apply our consideration to the long-living paraexcitons in Cu2O. Taking into account the exciton-phonon interaction and introducing a coherent phonon part of the moving condensate, we solve quasi-stationary equations for the exciton-phonon condensate. These equations support localized solutions, and we discuss the conditions for the inhomogeneous condensate to appear in the crystal. Allowable values of the characteristic width of ballistic condensates are estimated. The stability conditions of the moving condensate are analyzed by use of Landau arguments, and Landau critical parameters appear in the theory. It follows that, under certain conditions, the condensate can move through the crystal as a stable droplet. To separate the coherent and non-coherent parts of the exciton-phonon packet, we suggest to turn off the phonon wind by the changes in design of the 3D crystal and boundary conditions for the moving droplet.Comment: 13 pages, LaTeX, three eps figures are incorporated by epsf. submitted to Phys. Letters