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Well-posedness of the periodic nonlinear Schrödinger equation with concentrated nonlinearity

Jinyeop Lee, Andrew Rout

TL;DR

We address the well-posedness of the 1D periodic nonlinear Schrödinger equation with a concentrated nonlinearity on the torus, proving global energy-conserving mild solutions for data in $H^1(\mathbb{T})$ and a local theory below the energy space. The main approach combines two approximation schemes—the smoothed NLS with a scaled potential $V^{\varepsilon}$ and a concentrated complex Ginzburg–Landau equation—and an inviscid limit to connect them to the target equation, with a Volterra formulation for the charge at the origin driving the analysis. The paper contributes the first rigorous periodic theory for a concentrated NLS, including global energy conservation in $H^1(\mathbb{T})$, a local well-posedness result for $s\in(\tfrac12,1)$ via Wiener algebra, and a uniqueness mechanism based on the inviscid limit of the CGL model. This framework provides a robust foundation for point interactions on periodic domains and offers a blueprint for potential extension to higher dimensions and mixed nonlinearities.

Abstract

We study the solution theory of the nonlinear Schrödinger equation with a concentrated nonlinearity on the torus. In particular, we establish existence and uniqueness of global energy-conserving solutions for initial data in $H^1$. Our approach is based on two approximation schemes, namely the concentrated limit of a smoothed nonlinear Schrödinger equation and the inviscid limit of a concentrated complex Ginzburg--Landau equation. We also prove the existence and uniquness of solutions below the energy space. To our knowledge, this is the first rigorous solution theory for a periodic nonlinear Schrödinger equation with a concentrated nonlinearity.

Well-posedness of the periodic nonlinear Schrödinger equation with concentrated nonlinearity

TL;DR

We address the well-posedness of the 1D periodic nonlinear Schrödinger equation with a concentrated nonlinearity on the torus, proving global energy-conserving mild solutions for data in and a local theory below the energy space. The main approach combines two approximation schemes—the smoothed NLS with a scaled potential and a concentrated complex Ginzburg–Landau equation—and an inviscid limit to connect them to the target equation, with a Volterra formulation for the charge at the origin driving the analysis. The paper contributes the first rigorous periodic theory for a concentrated NLS, including global energy conservation in , a local well-posedness result for via Wiener algebra, and a uniqueness mechanism based on the inviscid limit of the CGL model. This framework provides a robust foundation for point interactions on periodic domains and offers a blueprint for potential extension to higher dimensions and mixed nonlinearities.

Abstract

We study the solution theory of the nonlinear Schrödinger equation with a concentrated nonlinearity on the torus. In particular, we establish existence and uniqueness of global energy-conserving solutions for initial data in . Our approach is based on two approximation schemes, namely the concentrated limit of a smoothed nonlinear Schrödinger equation and the inviscid limit of a concentrated complex Ginzburg--Landau equation. We also prove the existence and uniquness of solutions below the energy space. To our knowledge, this is the first rigorous solution theory for a periodic nonlinear Schrödinger equation with a concentrated nonlinearity.

Paper Structure

This paper contains 24 sections, 33 theorems, 151 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Method of Proof
  3. Previously Known Results
  4. Outline of the Paper
  5. Notation and Preliminaries
  6. Notation
  7. Preliminary Results
  8. Construction of Energy Conserving Solutions
  9. Uniform Bounds on the Smoothed NLS
  10. Existence of Solutions for the \ref{['concentrated_NLS_introduction']}
  11. Conservation Laws for a Solution of the \ref{['concentrated_NLS_introduction']}
  12. Analysis of the Complex Ginzburg--Landau Equations
  13. Uniqueness of Solutions
  14. Volterra Integral Equations
  15. Analysis of Convolution Kernels
  16. ...and 9 more sections

Key Result

Theorem 1.1

Suppose that $u_0 \in H^1(\mathbb{T})$. Then, for any $T > 0$, there is a unique function $u \in C([0,T]; H^1(\mathbb{T}))$ such that $u$ solves concentrated_NLS_introduction. Moreover, one has for any $t \in [0,T]$.

Figures (1)

  • Figure 1: Diagram illustrating the limits for $\gamma \to 0$ and $\varepsilon \to 0$

Theorems & Definitions (70)

  • Theorem 1.1
  • Remark 1.2
  • Remark 1.3
  • Remark 1.4
  • Theorem 1.5: Existence of unique mass-conserving solutions below $H^1$
  • Remark 1.6
  • Remark 1.7
  • Remark 1.8
  • Lemma 2.1
  • proof
  • ...and 60 more