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.