Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Gray and black solitons of nonlocal Gross-Pitaevskii equations: existence, monotonicity and nonlocal-to-local limit

André de Laire, Salvador López-Martínez

Abstract

This article investigates the qualitative aspects of dark solitons of one-dimensional Gross-Pitaevskii equations with general nonlocal interactions, which correspond to traveling waves with subsonic speeds. Under general conditions on the potential interaction term, we provide uniform bounds, demonstrate the existence of symmetric solitons, and identify conditions under which monotonicity is lost. Additionally, we present new properties of black solitons. Moreover, we establish the nonlocal-to-local convergence, i.e. the convergence of the soliton of the nonlocal model toward the explicit dark solitons of the local Gross-Pitaevskii equation.

Gray and black solitons of nonlocal Gross-Pitaevskii equations: existence, monotonicity and nonlocal-to-local limit

Abstract

This article investigates the qualitative aspects of dark solitons of one-dimensional Gross-Pitaevskii equations with general nonlocal interactions, which correspond to traveling waves with subsonic speeds. Under general conditions on the potential interaction term, we provide uniform bounds, demonstrate the existence of symmetric solitons, and identify conditions under which monotonicity is lost. Additionally, we present new properties of black solitons. Moreover, we establish the nonlocal-to-local convergence, i.e. the convergence of the soliton of the nonlocal model toward the explicit dark solitons of the local Gross-Pitaevskii equation.
Paper Structure (16 sections, 38 theorems, 251 equations, 2 figures)

This paper contains 16 sections, 38 theorems, 251 equations, 2 figures.

Table of Contents

  1. Introduction
  2. The problem
  3. Hypotheses and main results
  4. Existence results
  5. Nonlocal-to-local limit
  6. Existence of symmetric solitons for small lambda
  7. Monotonicity results
  8. Examples and applications
  9. A priori estimates
  10. Decay and analyticity of finite energy solutions
  11. Nonlocal-to-local limit for gray solitons
  12. Existence of symmetric solitons for small lambda
  13. Solitons with oscillations
  14. Analysis of black solitons
  15. Monotone black solitons
  16. ...and 1 more sections

Key Result

Theorem 1.1

Assume that Then, for almost every $c\in (0,\sqrt{2\mathfrak{s}})$, there exists a nontrivial solution $u\in \mathcal{E}(\mathbb{R})$ to TWc.

Figures (2)

  • Figure 1: Numerically computed solitons for potential \ref{['pot:Gauss']} with $\lambda=3.0$, showing $\eta$ (left panel) and ${\theta}_k$ (right panel) as function of $x$.
  • Figure 2: Numerically computed solitons for potential \ref{['pot:vander']} with $\beta=0.5$ and $c=0.1$ for several values of $\lambda$, plotted for $\eta=1-|u |^2.$

Theorems & Definitions (75)

  • Theorem 1.1: dLMar2022dLDuMar23
  • Theorem 1.2: Theorem 4.1 in dLMar2022
  • Corollary 1.3: dLMar2022
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Remark 1.7
  • Theorem 1.8
  • Theorem 1.9
  • Theorem 1.10
  • ...and 65 more