Dispersive shock waves theory with account of non-Kerr nonlinearity, weak dissipation and cylindrical geometry.

Abstract

We are going to consider nonlinear wave structures evolving from intensive initial pulses through formation of dispersive shock waves. These structures are observed in various media including water waves, Bose-Einstein condensates (BECs), plasma, and optical fibers. So far the theory of dispersive shock waves has been mainly developing for non-dissipative media when wave dynamics is described by completely integrable equations as, for example, Korteweg-de Vries equation or NLS equation. We will consider here generalization of the theory on situations that are described by non-integrable equations. First, we will discuss the problem of evolution of an initial discontinuity according to the modified Korteweg-de Vries equation, which can be obtained as an approximation to the system of Gross-Pitaevskii equations for two-component BEC, and in our approach we will take into account small Burgers viscosity. Second, we will develop the method of calculation of number of solitons produced from an initially intensive pulse when its evolution is described by the generalized NLS (Gross-Pitaevskii) equation with non-Kerr nonlinearity. Third, we will consider the role of cylindrical geometry in evolution of dispersive shock waves, and this problem is important for understanding the dynamics of polariton condensates. Solution of these problems will make important contribution into the topical subject of the modern nonlinear physics.