Axial Symmetry of Normalized Solutions for Magnetic Gross-Pitaevskii Equations with Anharmonic Potentials

TL;DR

This paper proves the existence and uniqueness of axially symmetric normalized solutions for magnetic Gross-Pitaevskii equations with anharmonic potentials in 2D and 3D, showing solutions are vortex-free in certain cases.

Contribution

It establishes the existence, axial symmetry, and uniqueness of normalized solutions for these equations, including the vortex-free property in non-radially symmetric potentials.

Findings

Existence of axially symmetric solutions as parameter approaches critical value.

Uniqueness of solutions up to phase and rotation transformations.

Vortex-free solutions in the 3D case with non-radially symmetric potentials.

Abstract

This paper is concerned with normalized solutions of the magnetic focusing Gross-Pitaevskii equations with anharmonic potentials in $\R^N$, where $N=2,3$. The existence of axially symmetric solutions is constructed as the parameter $a>0$ satisfies $a \to a_*(N)$, where $a_*(N)\geq0$ is a critical constant depending only on $N$. We further prove that up to the constant phase and rotational transformation, normalized concentrating solutions as $a\to a_*(N)$ must be unique and axially symmetric. As a byproduct, we also obtain that for the case $N=3$, the normalized concentrating solution as $a\to a_*(3)$ is free of vortices, where the anharmonic potential is non-radially symmetric.