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 $\bar{\partial }$ generalization of the nonlinear steepest descent method we compute the long-time asymptotic expansion of the solution $\psi (x,t)$ in any fixed space-time cone $C(x_1,x_2,v_1,v_2)=\left\{(x,t)\in {\mathbb{R}}^2:x=x_0+vt\text{ with }{x}_0\in [x_1,x_2],v\in [v_1,v_2]\right\}$ up to an (optimal) residual error of order $\mathcal{O}\left(t^{-3/4}\right)$. 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 $L^2(\mathbb{R})$ moment and (weak) derivative and that it not generate any spectral singularities.