Long time behavior for the focusing nonlinear schroedinger equation with real spectral singularities

Abstract

We consider the effect of real spectral singularities on the long time behavior of the solutions of the focusing Nonlinear Schroedinger equation. We find that for each spectral singularity λ′ ∈ℝ, such an effect is limited to the region of the (x,t)-plane in which λ′ is close to the point of stationary phase$\lambda _0 = \tfrac{{ - x}}{{4t}}$ (the phase here being defined in a standard way by, say, the evolution of the Jost functions). In that region, the solution performs decaying oscillations of the same form as in the other regions, but with different parameters. The order of decay is$O\left( {\left( {\tfrac{{\log t}}{t}} \right)^{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}} } \right)$.

We prove our result by using the Riemann-Hilbert factorization formulation of the inverse scattering problem. We recover our asymptotics by transforming our problem to one which is equivalent for large time, and which can be interpreted as the one corresponding to the genus 0 algebro-geometric solution of the equation.