Long time asymptotics for the focusing nonlinear Schr\"odinger equation in the solitonic region with the presence of high-order discrete spectrum

TL;DR

This paper analyzes the long-time behavior of solutions to the focusing nonlinear Schrödinger equation with high-order discrete spectrum using the $ar{ ext{D}}$ steepest descent method, confirming the soliton resolution conjecture in this context.

Contribution

It develops a novel approach to characterize high-order poles and formulates an enlarged RH problem, leading to precise asymptotics for solutions with high-order discrete spectrum.

Findings

Solution satisfies the soliton resolution conjecture.

Asymptotic expansion includes high-order pole-solitons.

Error term is of order $ ext{O}(t^{-3/4})$ from the $ar{ ext{D}}$ equation.

Abstract

In this paper, we use the $\bar{\partial}$ steepest descent method to study the initial value problem for focusing nonlinear Schr\"odinger (fNLS) equation with non-generic weighted Sobolev initial data that allows for the presence of high-order discrete spectrum. More precisely, we shall characterize the properties of the eigenfunctions and scattering coefficients in the presence of high-order poles; further we formulate an appropriate enlarged RH problem; after a series of deformations, the RH problem is transformed into a solvable model. Finally, we obtain the asymptotic expansion of the solution of the fNLS equation in any fixed space-time cone: %as $t \to \infty$, \begin{equation*} \mathcal{S}(x_1,x_2,v_1,v_2):=\left\lbrace (x,t)\in \mathbb{R}^2: x=x_0+vt, \ x_0\in[x_1,x_2]\text{, }v\in[v_1,v_2]\right\rbrace. \end{equation*} Observing the result indicates that the solution of fNLS equation in this case satisfies the soliton resolution conjecture. 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 of this result is up to $\mathcal{O}(t^{-3/4})$ which comes from the corresponding $\bar{\partial}$ equation.