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Momentum-mass normalized dark-bright solitons to one dimensional Gross-Pitaevskii systems

Salvador López-Martínez

TL;DR

The paper proves the existence of dark-bright solitons as traveling-wave solutions to a one-dimensional defocusing Gross-Pitaevskii system in the miscibility regime α^2 ≤ β. By formulating a constrained variational problem for the renormalized energy with two constraints (renormalized momentum p(u)=q and bright component mass ||v||_2^2 = m), and employing symmetric decreasing rearrangements with concentration-compactness, it establishes the existence of a minimizer (u,v) that is symmetric and strictly positive for the bright component and strictly inside the unit disk for the dark component. The minimizer solves a traveling-wave system with explicit constraints, yielding subsonic speed (0<c<√2) and detailed information on the Lagrange multipliers c and λ. The approach highlights the role of the miscibility condition in ensuring nonnegativity and compactness, and provides a rigorous variational construction of nontrivial dark-bright solitons beyond integrable cases.

Abstract

We rigorously establish the existence of dark-bright solitons as traveling wave solutions to a one dimensional defocusing Gross-Pitaevskii system, a widely used model for describing mixtures of Bose-Einstein condensates and nonlinear optical systems. These solitons are shown to exhibit symmetry and radial monotonicity in modulus, and to propagate at subsonic speed. Our method relies on minimizing an energy functional subject to two constraints: the mass of the bright component and a modified momentum of the dark component. The compactness of minimizing sequences is obtained via a concentration-compactness argument, which requires some novel estimates based on symmetric decreasing rearrangements.

Momentum-mass normalized dark-bright solitons to one dimensional Gross-Pitaevskii systems

TL;DR

The paper proves the existence of dark-bright solitons as traveling-wave solutions to a one-dimensional defocusing Gross-Pitaevskii system in the miscibility regime α^2 ≤ β. By formulating a constrained variational problem for the renormalized energy with two constraints (renormalized momentum p(u)=q and bright component mass ||v||_2^2 = m), and employing symmetric decreasing rearrangements with concentration-compactness, it establishes the existence of a minimizer (u,v) that is symmetric and strictly positive for the bright component and strictly inside the unit disk for the dark component. The minimizer solves a traveling-wave system with explicit constraints, yielding subsonic speed (0<c<√2) and detailed information on the Lagrange multipliers c and λ. The approach highlights the role of the miscibility condition in ensuring nonnegativity and compactness, and provides a rigorous variational construction of nontrivial dark-bright solitons beyond integrable cases.

Abstract

We rigorously establish the existence of dark-bright solitons as traveling wave solutions to a one dimensional defocusing Gross-Pitaevskii system, a widely used model for describing mixtures of Bose-Einstein condensates and nonlinear optical systems. These solitons are shown to exhibit symmetry and radial monotonicity in modulus, and to propagate at subsonic speed. Our method relies on minimizing an energy functional subject to two constraints: the mass of the bright component and a modified momentum of the dark component. The compactness of minimizing sequences is obtained via a concentration-compactness argument, which requires some novel estimates based on symmetric decreasing rearrangements.

Paper Structure

This paper contains 6 sections, 16 theorems, 180 equations.

Table of Contents

  1. Introduction
  2. Functional setting and main results
  3. Basics of symmetric decreasing rearrangements
  4. Properties of the minimizing surface
  5. Compactness of minimizing sequences
  6. Proof of the main result

Key Result

Theorem 2.1

For every $q\in (0,\pi/2)$ there exists $\mathbf{u}_q\in\mathcal{NE}(\mathbb{R})$, with $0<|\mathbf{u}_q|<1$, $p(\mathbf{u}_q)=q$, such that Moreover, there exists $c=c(q)\in (0,\sqrt{2})$ such that $E(\mathbf{u}_q,0)=(2-c^2)^\frac{3}{2}/3$, and $\mathbf{u}_q$ is the unique solution, up to invariances, to eq:u. These solutions are explicit and are given by Besides, there exists $\mathbf{u}_{\pi/

Theorems & Definitions (34)

  • Theorem 2.1: bethuel0
  • Remark 2.2
  • Theorem 2.3
  • Theorem 2.4
  • Definition 3.1
  • Proposition 3.2
  • Proposition 3.3
  • Theorem 3.4
  • Theorem 3.5
  • Theorem 3.6
  • ...and 24 more