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

TL;DR

This paper proves the existence of dark-bright solitons in a one-dimensional Gross-Pitaevskii system, revealing their symmetry, monotonicity, and subsonic propagation through a novel variational approach.

Contribution

It introduces a new variational method with constraints to rigorously establish dark-bright solitons in the system, including novel estimates for compactness.

Findings

Dark-bright solitons exist as traveling wave solutions.

Solitons exhibit symmetry and radial monotonicity.

Propagation occurs at subsonic speeds.

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.