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Global well-posedness for the generalized intermediate NLS with a nonvanishing condition at infinity

Takafumi Akahori, Rana Badreddine, Slim Ibrahim, Nobu Kishimoto

TL;DR

The paper addresses global well-posedness for the intermediate nonlinear Schrödinger equation with nonvanishing boundary data, focusing on dark-soliton-type solutions. It develops a Zhidkov-type functional framework \mathcal{Z}^2_\rho tailored to nonzero boundaries and employs a modified energy method to control derivative nonlinearities and the singular operator  T_\delta . The authors establish local well-posedness in \mathcal{Z}^2_\rho, then derive two conservation laws \mathcal{H}_1(u)  and \mathcal{H}_2(u)  compatible with nonvanishing data to obtain global control, proving global well-posedness for the generalized INLS and providing uniform bounds in the integrable case. This work extends well-posedness theory to nonvanishing boundary conditions and dark solitons, without relying on complete integrability, and has implications for the study of internal-wave models with finite-depth effects.

Abstract

The Intermediate Nonlinear Schrödinger equation models quasi-harmonic internal waves in two-fluid layer system, and admits dark solitons, that is, solutions with nonvanishing boundary conditions at spatial infinity. These solutions fall outside existing well-posedness theories. We establish local and global well-posedness in a Zhidkov-type space naturally suited to such non-trivial boundary conditions, and extend these results to a generalized defocusing equation. This appears to be the first well-posedness result for the equation in a functional setting adapted to its dark soliton structure.

Global well-posedness for the generalized intermediate NLS with a nonvanishing condition at infinity

TL;DR

The paper addresses global well-posedness for the intermediate nonlinear Schrödinger equation with nonvanishing boundary data, focusing on dark-soliton-type solutions. It develops a Zhidkov-type functional framework \mathcal{Z}^2_\rho tailored to nonzero boundaries and employs a modified energy method to control derivative nonlinearities and the singular operator  T_\delta . The authors establish local well-posedness in \mathcal{Z}^2_\rho, then derive two conservation laws \mathcal{H}_1(u)  and \mathcal{H}_2(u)  compatible with nonvanishing data to obtain global control, proving global well-posedness for the generalized INLS and providing uniform bounds in the integrable case. This work extends well-posedness theory to nonvanishing boundary conditions and dark solitons, without relying on complete integrability, and has implications for the study of internal-wave models with finite-depth effects.

Abstract

The Intermediate Nonlinear Schrödinger equation models quasi-harmonic internal waves in two-fluid layer system, and admits dark solitons, that is, solutions with nonvanishing boundary conditions at spatial infinity. These solutions fall outside existing well-posedness theories. We establish local and global well-posedness in a Zhidkov-type space naturally suited to such non-trivial boundary conditions, and extend these results to a generalized defocusing equation. This appears to be the first well-posedness result for the equation in a functional setting adapted to its dark soliton structure.

Paper Structure

This paper contains 11 sections, 14 theorems, 182 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notation
  4. Properties of $T_{\delta}$
  5. Properties of the space $\mathcal{Z}^k_\rho$ and control of Littlewood-Paley operators
  6. Estimates for the free Schrödinger operator
  7. Nonlinear estimates and an integration by parts lemma
  8. Local well-posedness
  9. Global well-posedness
  10. Fourier transform of the hyperbolic cotangent function
  11. Proof of Lemma \ref{['lem:fe']} and \ref{['est:fe']}

Key Result

Theorem 1.1

Let $\delta>0$, $\alpha,\beta\in \mathbb{R}$ be arbitrary. For $M_0>0$, there exists $T_*\in (0,1]$ satisfying $T_*\geq c_{\alpha,\beta}(1+\delta^{-1})^{-1}(1+M_0^4)^{-1}$ such that the following holds. For any $\rho>0$ and $\phi \in \mathcal{Z}^2_\rho$ with $\mathcal{E}^2_\rho(\phi)\leq M_0^2$, the

Theorems & Definitions (31)

  • Theorem 1.1: Local well-posedness in $\mathcal{Z}^2_\rho$
  • Remark 1.1
  • Theorem 1.2: Global well-posedness and a priori bound
  • Theorem 1.2: Global well-posedness and a priori bound
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • Remark 2.1
  • proof : Proof of Lemma \ref{['lem:ibp']}
  • Proposition 3.1
  • ...and 21 more