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The defocusing Calogero--Moser derivative nonlinear Schr{ö}dinger equation with a nonvanishing condition at infinity

Xi Chen

TL;DR

This work analyzes the defocusing Calogero-Moser derivative NLS on the real line with nonvanishing boundary at infinity. It develops a functional framework based on the energy space $E$ and its chiral subspace $E_+$, proves global well-posedness in $E$, and establishes uniform long-time bounds via a Lax-pair structure and two conservation laws. A key contribution is an explicit Poisson-type formula for chiral solutions in $E_+$, extending known explicit formulas from $H_+^s$ to the energy setting. The combination of local well-posedness, conserved quantities, and the explicit solution representation provides tools to study long-time dynamics, potential zero-dispersion limits, and traveling waves within this nonzero-background regime.

Abstract

We consider the defocusing Calogero--Moser derivative nonlinear Schr{ö}dinger equation\begin{align*}i \partial_{t} u+\partial_{x}^2 u-2ΠD\left(|u|^{2}\right)u=0, \quad (t,x ) \in \mathbb{R} \times \mathbb{R}\end{align*}posed on $E := \left\{u \in L^{\infty}(\mathbb{R}): u' \in L^{2}(\mathbb{R}), u'' \in L^{2}(\mathbb{R}), |u|^{2}-1 \in L^{2}(\mathbb{R})\right\}$. We prove the global well-posedness of this equation in $E$. Moreover, we give an explicit formula for the chiral solution to this equation.

The defocusing Calogero--Moser derivative nonlinear Schr{ö}dinger equation with a nonvanishing condition at infinity

TL;DR

This work analyzes the defocusing Calogero-Moser derivative NLS on the real line with nonvanishing boundary at infinity. It develops a functional framework based on the energy space and its chiral subspace , proves global well-posedness in , and establishes uniform long-time bounds via a Lax-pair structure and two conservation laws. A key contribution is an explicit Poisson-type formula for chiral solutions in , extending known explicit formulas from to the energy setting. The combination of local well-posedness, conserved quantities, and the explicit solution representation provides tools to study long-time dynamics, potential zero-dispersion limits, and traveling waves within this nonzero-background regime.

Abstract

We consider the defocusing Calogero--Moser derivative nonlinear Schr{ö}dinger equation\begin{align*}i \partial_{t} u+\partial_{x}^2 u-2ΠD\left(|u|^{2}\right)u=0, \quad (t,x ) \in \mathbb{R} \times \mathbb{R}\end{align*}posed on . We prove the global well-posedness of this equation in . Moreover, we give an explicit formula for the chiral solution to this equation.

Paper Structure

This paper contains 11 sections, 12 theorems, 135 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. Preliminaries
  4. The space $E$ and the Sobolev spaces
  5. Action of the linear Schrödinger group on the space $E$
  6. The Lax pair
  7. Well-posedness and conservation laws
  8. Local well-posedness
  9. Conservation laws
  10. Global well-posedness
  11. Explicit formula for chiral solutions

Key Result

Theorem 1.1

Given $u_0 \in E$, there exists a unique solution $u \in C\left(\mathbb{R}; E\right)$ of (1.1) with the initial data $u(0) = u_0$, and this global solution satisfies $\sup_{t \in \mathbb{R}} \|u(t)\|_{X^2} < +\infty$ and $\sup_{t \in \mathbb{R}} \||u(t)|^2-1\|_{L^2} < +\infty$. Furthermore, for ever

Theorems & Definitions (26)

  • Theorem 1.1
  • Theorem 1.2
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • ...and 16 more