Pinwheel solutions to Schr\"odinger systems

TL;DR

This paper proves the existence of rotationally related positive solutions in competitive Schr"odinger systems with trapping potentials, and explores their behavior under varying interaction strengths, leading to optimal partitions of the domain.

Contribution

It introduces a novel class of solutions where each component is a rotation of the previous one, and analyzes their asymptotic behavior in different interaction regimes.

Findings

Existence of positive segregated solutions with rotational symmetry.

Behavior characterization as interaction forces tend to zero or infinity.

Derivation of optimal partitions linearly isometric to each other.

Abstract

We establish the existence of positive segregated solutions for competitive nonlinear Schr\"odinger systems in the presence of an external trapping potential, which have the property that each component is obtained from the previous one by a rotation, and we study their behavior as the forces of interaction become very small or very large. As a consequence, we obtain optimal partitions for the Schr\"odinger equation by sets that are linearly isometric to each other.