Near soliton evolution for $2$-equivariant Schr\"odinger Maps in two space dimensions

TL;DR

This paper studies the evolution of equivariant solutions to the Schr"odinger Map equation in two dimensions, proving stability and dispersive bounds for small perturbations around the ground state in the challenging case of equivariance class |m|=2.

Contribution

It establishes the stability and global dispersive behavior of solutions near the ground state for the critical equivariance class |m|=2, including bounds on the modulation parameter growth.

Findings

Small perturbations lead to global solutions with dispersive decay.

The ground state is stable in a topology stronger than .

Solutions can drift along the soliton family over time.

Abstract

We consider equivariant solutions for the Schr\"odinger Map equation in $2+1$ dimensions, with values into $\mathbb{S}^2$. Within each equivariance class $m \in \mathbb{Z}$ this admits a lowest energy nontrivial steady state $Q^m$, which extends to a two dimensional family of steady states by scaling and rotation. If $|m| \geq 3$ then these ground states are known to be stable in the energy space $\dot H^1$, whereas instability and even finite time blow-up along the ground state family may occur if $|m| = 1$. In this article we consider the most delicate case $|m| = 2$. Our main result asserts that small $\dot H^1$ perturbations of the ground state $Q^2$ yield global in time solutions, which satisfy global dispersive bounds. Unlike the higher equivariance classes, here we expect solutions to move arbitrarily far along the soliton family; however, we are able to provide a time dependent bound on the growth of the scale modulation parameter. We also show that within the equivariant class the ground state is stable in a slightly stronger topology $X \subset \dot H^1$.