Uniqueness of Single Peak Solutions for Coupled Nonlinear Gross-Pitaevskii Equations with Potentials

TL;DR

This paper proves the uniqueness of single peak solutions in coupled nonlinear Gross-Pitaevskii equations under specific potential conditions, including radially symmetric non-monotonic potentials, and establishes symmetry and uniqueness of ground states.

Contribution

It provides the first uniqueness results for ground states with radially symmetric, non-monotonic potentials, extending previous work on solution uniqueness.

Findings

Single peak solutions are unique if they concentrate on the same point with similar potential Taylor expansions.

Positive ground states with radially symmetric, non-monotonic potentials are cylindrically symmetric and unique up to rotations.

The results include solutions in degenerate, ring-shaped potentials with minimal spheres.

Abstract

For a couple of singularly perturbed Gross-Pitaevskii equations, we first prove that the single peak solutions, if they concentrate on the same point, are unique provided that the Taylor's expansion of potentials around the concentration point is in the same order along all directions. Among other assumptions, our results indicate that the peak solutions obtained in [21,31,38] are unique. Moreover, for the radially symmetric ring-shaped potential, which attains its minimum at the spheres$\Gamma_i:=\{x\in\mathbb{R}^N:|x|=A_i>0\},i=1,2,\cdots,l,$ and is totally degenerate in the tangential space of $\Gamma_i$, we prove that the positive ground state is cylindrically symmetric and is unique up to rotations around the origin. Aa far as we know, this is the first uniqueness result for ground states under radially symmetric but non-monotonic potentials.