Long time asymptotic behavior of the focusing nonlinear Schrödinger equation

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.