Bifurcating solitonic vortices in a strip

TL;DR

This paper investigates bifurcations of solitonic vortices in a strip geometry, showing stationary solutions to the Gross-Pitaevskii equation with multiple vortices emerging as the strip width increases.

Contribution

It introduces a novel analysis of bifurcating solitonic vortex solutions in a strip, using Fourier decomposition and fixed point methods around the linearized operator.

Findings

Existence of stationary solutions with k vortices on a transverse line.

Bifurcation from soliton solutions as strip width increases.

Application of fixed point and inverse function theorem techniques.

Abstract

The specific geometry of a strip provides connections between solitons and solitonic vortices, which are vortices with a solitonic behaviour in the infinite direction of the strip. We show that there exist stationary solutions to the Gross-Pitaevskii equation with k vortices on a transverse line, which bifurcate from the soliton solution as the width of the strip is increased. After decomposing into Fourier series with respect to the transverse variable, the construction of these solitonic vortices is achieved by relying on a careful analysis of the linearized operator around the soliton solution: we apply a fixed point argument to solve the equation in the directions orthogonal to the kernel of the linearized operator, and then handle the direction corresponding to the kernel by an inverse function theorem.