Long time asymptotic behavior of the focusing nonlinear Schrödinger equation

Abstract

We study the Cauchy problem for the focusing nonlinear Schrödinger (fNLS) equation. Using the ∂‾ generalization of the nonlinear steepest descent method we compute the long-time asymptotic expansion of the solution ψ(x,t) in any fixed space-time cone C(x1,x2,v1,v2)={(x,t)∈R2:x=x0+vt with x0∈[x1,x2],v∈[v1,v2]} up to an (optimal) residual error of order O(t−3/4). In each cone C the leading order term in this expansion is a multi-soliton whose parameters are modulated by soliton–soliton and soliton–radiation interactions as one moves through the cone. Our results require that the initial data possess one L2(R) moment and (weak) derivative and that it not generate any spectral singularities.