Multiplicity of solutions for Gross-Pitaevskii equations on Riemannian manifolds

TL;DR

This paper establishes multiple solutions for the Gross-Pitaevskii equations on Riemannian manifolds, linking solution multiplicity to the topology of velocity sets using advanced variational and geometric methods.

Contribution

It introduces new multiplicity results for solutions of Gross-Pitaevskii equations on manifolds, utilizing critical point theory, $ ext{Gamma}$-convergence, and novel isoperimetric problem insights.

Findings

Lower bounds on solution multiplicity based on topology

Solutions characterized as critical points with prescribed momentum

New results for codimension 2 isoperimetric problems

Abstract

We provide a multiplicity result for solutions of time-independent Gross-Pitaevskii equations on closed Riemannian manifolds. Such solutions arise as (possibly non-minimizing) critical points of the Ginzburg-Landau energy having prescribed momentum according to a given tangent velocity field. Lower bounds on the multiplicity of solutions are obtained in terms of the topology of the maximum velocity set, in the small momentum and vorticity core size regime. The proof relies on methods from critical point theory and $\Gamma$-convergence for Ginzburg-Landau functionals as well as on some new results for codimension 2 isoperimetric-type problems in the small flux regime, possibly of independent interest.