Oscillatory Solutions for Soliton Equations; Phase Transitions and Variational Problems

Consider the Cauchy problem for the two following soliton equations: Korteweg-de-Vries with a small dispersion parameter epsilon and the nonlinear Schroedinger equation in 1+1 dimensions with cubic nonlinearity in the so-called semiclassical limit. It is known that at a certain "caustic" there is a "phase transition" in the solution of each of these equations. Oscillations of high frequency appear, so strong limits no longer exist. Soliton theory enables us to provide detailed information for the solutions as the small parameter epsilon goes to zero, prove the existence of weak limits, and describe completely the different oscillatory regions in terms of a variational problem of "electrostatic type". In the case of the focusing NLS this is a non-convex problem.