Multisoliton solutions for equivariant wave maps on a $2+1$ dimensional wormhole

TL;DR

This paper investigates multisoliton solutions in equivariant wave maps from a 2+1 dimensional wormhole to the 2-sphere, proposing a model for chains of kinks and anti-kinks with increasing expansion rates, validated by PDE simulations.

Contribution

It introduces a finite-dimensional ODE model for multisoliton dynamics in wave maps on a wormhole, supported by PDE verification.

Findings

Predicted existence of asymptotically static kink-anti-kink chains.

Effective ODE models accurately describe multisoliton dynamics.

Chains are at the threshold of kink-anti-kink annihilation.

Abstract

We study equivariant wave maps from the $2+1$ dimensional wormhole to the 2-sphere. This model has explicit harmonic map solutions which, in suitable coordinates, have the form of the sine-Gordon kinks/anti-kinks. We conjecture that there exist asymptotically static chains of $N\geq 2$ alternating kinks and anti-kinks whose subsequent rates of expansion increase in geometric progression as $t\rightarrow \infty$. Our argument employs the method of collective coordinates to derive effective finite-dimensional ODE models for the asymptotic dynamics of $N$-chains. For $N=2,3$ the predictions of these effective models are verified by direct PDE computations which demonstrate that the $N$-chains lie at the threshold of kink-anti-kink annihilation.