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 // e IR, such an effect is limited to the region of the (x,t)-plane in which /! is close to the point of stationary phase λo = ( 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((^) 12 )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

Fast Soliton Scattering by Delta Impurities

“…In the intermediate case of a zero resonance (a spectral singularity on the real axis) we apply the result of [10] to obtain…”

Soliton scattering by delta impurities

“…Here t q (v) and r q (v) are the transmission and reflection coefficients of the deltapotential (see (10)…”

“…The use of the Strichartz estimates as an approximation device, as opposed to say energy estimates, is critical since the estimates obtained depend only upon the L 2 norm of the solution, which is conserved and independent of v. Thus, v functions as an asymptotic parameter; larger v means a shorter interaction phase and a better approximation of the solution by the linear flow. Theorem 2 combines this analysis with the inverse scattering method [11], [4], [3], [10]. The delta potential splits the incoming soliton into two waves which become single solitons.…”

On Universality of Critical Behavior in the Focusing Nonlinear Schrödinger Equation, Elliptic Umbilic Catastrophe and the Tritronquée Solution to the Painlevé-I Equation

“…Properties of various classes of solutions to this equation have been extensively studied both analytically and numerically (Bronski and Kutz 2002;Buckingham and Venakides 2007;Carles 2007;Ceniceros and Tian 2002;Forest and Lee 1986;Grenier 1998;Kamvissis 1996;Kamvissis et al 2003;Klein 2006;Lyng and Miller 2007;Miller and Kamvissis 1998;Tovbis et al 2004Tovbis et al , 2006. One of the striking features that distinguishes this equation from, say, the defocusing case…”