Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Multiplicity of solutions for Gross-Pitaevskii equations on Riemannian manifolds

Dario Corona, Stefano Nardulli, Ramon Oliver-Bonafoux, Giandomenico Orlandi

TL;DR

This work proves a multiplicity result for time-independent Gross–Pitaevskii equations on closed Riemannian manifolds under a momentum constraint along a solenoidal vector field $X$ that is co-exact. The authors formulate a constrained Ginzburg–Landau energy $E_{oldsymbol{ u},oldsymbol{ heta}}$ and, for small momentum $oldsymbol{ heta}$ and vortex core size $oldsymbol{ u}$, show that the number of critical points is bounded below by the Lusternik–Schnirelmann category of the maximum-velocity set $oldsymbol{ abla}oldsymbol{oldsymbol{ extSigma}}$. The core of the method is the photography technique, which builds a map from $oldsymbol{ abla}oldsymbol{oldsymbol{ extSigma}}$ to low-energy configurations and a barycenter that retracts back, leveraging $oldsymbol{ u}$-Γ-convergence of the energy to a codimension-2 isoperimetric problem constrained by flux. The results also establish energy concentration near $oldsymbol{ abla}oldsymbol{oldSigma}$ and connect the vortex geometry to minimal “isoperimetric submanifolds” in codimension two, contributing to the broader study of vortices in Ginzburg–Landau-type theories on manifolds with geometric constraints.

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 $Γ$-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.

Multiplicity of solutions for Gross-Pitaevskii equations on Riemannian manifolds

TL;DR

This work proves a multiplicity result for time-independent Gross–Pitaevskii equations on closed Riemannian manifolds under a momentum constraint along a solenoidal vector field that is co-exact. The authors formulate a constrained Ginzburg–Landau energy and, for small momentum and vortex core size , show that the number of critical points is bounded below by the Lusternik–Schnirelmann category of the maximum-velocity set . The core of the method is the photography technique, which builds a map from to low-energy configurations and a barycenter that retracts back, leveraging -Γ-convergence of the energy to a codimension-2 isoperimetric problem constrained by flux. The results also establish energy concentration near and connect the vortex geometry to minimal “isoperimetric submanifolds” in codimension two, contributing to the broader study of vortices in Ginzburg–Landau-type theories on manifolds with geometric constraints.

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 -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.

Paper Structure

This paper contains 11 sections, 23 theorems, 232 equations, 2 figures.

Table of Contents

  1. Introduction and Main Result
  2. Notation and Preliminaries
  3. General notions of Geometric Measure Theory
  4. Our Setting
  5. Outline of the proof
  6. Functional setting
  7. --convergence
  8. Photography map
  9. Barycenter map
  10. Proof of Theorem \ref{['theorem:main']}
  11. On the behavior of sequences of almost minimizers of the isoperimetric problems

Key Result

Theorem 1

Let $(M,g)$ be a closed Riemannian manifold of dimension $N \ge 3$, and $X\colon M \to TM$ a smooth vector field such that its associated one-form is co-exact and ass:Sigmadelta-retraction holds. Let $W\colon \mathbb{C} \to [0,+\infty)$ be a $C^2$ potential with $W^{-1}(0)=\mathbb{S}^1$ that satisfi

Figures (2)

  • Figure 1: A small geodesic disk $D_X(p,r)$ orthogonal to $X$ in a $3$--dimensional ambient manifold $M$.
  • Figure 2: Construction of the photography map: using a cylindrical coordinate system in a local chart, we define $f_{\varepsilon,\phi}(p)$ by transporting the dipole vortex map $\omega_{p,\phi}$ onto $M$. The singularity, located at a distance $r(p,\phi)$, is shifted to the boundary $\partial D_X(p,r(p,\phi))$ and then removed by a linear smoothing.

Theorems & Definitions (54)

  • Theorem 1: Main result
  • Remark 1.1: Existence of minimizers
  • Remark 1.2: Further properties of the solutions
  • Remark 1.3: On the dependence of the critical parameters on $W$
  • Remark 1.4: Existence of admissible vector fields for arbitrary manifolds
  • Remark 1.5: The case of manifolds with trivial first homology group
  • Remark 1.6: On the case $\Sigma=M$
  • Remark 1.7: On the non-degeneracy condition for solutions
  • Remark 2.1: On $J_M(\phi,X)$ and the co-exactness assumption on $X$
  • Theorem 3.1: Photography method, cf. Benci1995BenciCerami1994
  • ...and 44 more