Spiraling solutions of nonlinear Schr\"odinger equations

TL;DR

This paper introduces a new family of sign-changing, non-axially symmetric solutions to the nonlinear Schrödinger equation that exhibit a spiraling symmetry akin to a helicoid, expanding understanding of solution structures.

Contribution

It establishes the existence of spiraling solutions with specific symmetry properties and analyzes how their shape varies with parameters, differing from prior Allen-Cahn studies.

Findings

Existence of non-axially symmetric, spiraling solutions

Shape of solutions depends on rotational slope parameter

Solutions exhibit invariance under screw motion

Abstract

We study a new family of sign-changing solutions to the stationary nonlinear Schr\"odinger equation $$ -\Delta v +q v =|v|^{p-2} v, \qquad \text{in $\mathbb{R}^3$,} $$ with $2<p<\infty$ and $q \ge 0$. These solutions are spiraling in the sense that they are not axially symmetric but invariant under screw motion, i.e., they share the symmetry properties of a helicoid. In addition to existence results, we provide information on the shape of spiraling solutions, which depends on the parameter value representing the rotational slope of the underlying screw motion. Our results complement a related analysis of Del Pino, Musso and Pacard for the Allen-Cahn equation, whereas the nature of results and the underlying variational structure are completely different.