The focusing NLS equation with step-like oscillating background: Asymptotics in a transition zone

IF 2.3 2区数学 Q1 MATHEMATICS

Journal of Differential Equations Pub Date : 2025-02-26 DOI:10.1016/j.jde.2025.02.016

Anne Boutet de Monvel , Jonatan Lenells , Dmitry Shepelsky

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引用次数: 0

Abstract

In a recent paper, we presented scenarios of long-time asymptotics for the solution of the focusing nonlinear Schrödinger equation with initial data approaching plane waves of the form

A_{1} e^{i ϕ_{1}} e^{- 2 i B_{1} x}

and

A_{2} e^{i ϕ_{2}} e^{- 2 i B_{2} x}

at minus and plus infinity, respectively. In the shock case

B_{1} < B_{2}

some scenarios include sectors of genus 3, that is, sectors

ξ_{1} < ξ < ξ_{2}

ξ ≔ x / t

, where the leading term of the asymptotics is expressed in terms of hyperelliptic functions attached to a Riemann surface of genus 3. The present paper deals with the asymptotic analysis in a transition zone between two genus 3 sectors. The leading term is expressed in terms of elliptic functions attached to a Riemann surface of genus 1. A central step in the derivation is the construction of a local parametrix in a neighborhood of two merging branch points.

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具有阶跃振荡背景的聚焦NLS方程：过渡区内的渐近性

在最近的一篇论文中，我们给出了聚焦非线性Schrödinger方程的长时间渐近解的场景，初始数据分别接近形式为A1eiϕ1e - 2iB1x和A2eiϕ2e - 2iB2x的平面波在负无穷和正无穷处。在冲击情况B1<；B2中，某些场景包含3属的扇区，即扇区ξ1<；ξ<ξ2， ξ是x/t，其中渐近的首项用附属于3属的黎曼曲面的超椭圆函数表示。本文讨论了两个3属扇区之间的过渡区内的渐近分析。首项是用附在黎曼曲面上的椭圆函数来表示的。推导的中心步骤是在两个合并分支点的邻域中构造局部参数。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Journal of Differential Equations 数学-数学

CiteScore

4.40

自引率

8.30%

发文量

543

审稿时长

9 months

期刊介绍： The Journal of Differential Equations is concerned with the theory and the application of differential equations. The articles published are addressed not only to mathematicians but also to those engineers, physicists, and other scientists for whom differential equations are valuable research tools. Research Areas Include: • Mathematical control theory • Ordinary differential equations • Partial differential equations • Stochastic differential equations • Topological dynamics • Related topics