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

TL;DR

This paper studies the existence, properties, and limits of dark solitons in nonlocal Gross-Pitaevskii equations, revealing conditions for symmetry, monotonicity, and convergence to local models.

Contribution

It provides new theoretical results on symmetric and black solitons, their monotonicity loss, and the nonlocal-to-local limit for one-dimensional nonlocal Gross-Pitaevskii equations.

Findings

Existence of symmetric dark solitons under general conditions

Identification of conditions where monotonicity is lost

Proof of convergence from nonlocal to local dark solitons

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.