Long time asymptotic behavior of the focusing nonlinear Schrodinger equation

We study the Cauchy problem for the focusing nonlinear Schrodinger (fNLS) equation. Using the (partial derivative)over-bar 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)) = {(x, t) is an element of R-2 : x = x(0) + v(t) with x(0) is an element of [x(1), x(2)], v is an element of [v(1), v(2)]} 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 L-2(R) moment and (weak) derivative and that it not generate any spectral singularities. (C) 2017 Elsevier Masson SAS. All rights reserved.