Long time asymptotics for the focusing nonlinear Schrödinger equation in the solitonic region with the presence of high-order discrete spectrum

Abstract

In this paper, we study the initial value problem for focusing nonlinear Schrödinger (fNLS) equation with non-generic weighted Sobolev initial data that allows for the presence of high-order discrete spectrum. More precisely, we show how to characterize the properties of the eigenfunctions and scattering coefficients in the presence of high-order poles; Further the initial value problem is formulated into an appropriate enlarged RH problem, which is transformed into a solvable model after a series of deformations. Finally, we obtain the asymptotic expansion of the solution of the fNLS equation in any fixed space-time cone:S(x1,x2,v1,v2):={(x,t)∈R2:x=x0+vt,x0∈[x1,x2], v∈[v1,v2]}. Our result is a verification of the soliton resolution conjecture for the fNLS equation in the solitonic region with the presence of high-order discrete spectrum. The leading order term of this solution includes a high-order pole-soliton whose parameters are affected by soliton-soliton interactions through the cone and soliton-radiation interactions on continuous spectrum. The error term is up to O(t−3/4) which comes from the corresponding ∂¯ equation.