Equivariant Wave Maps on the Hyperbolic Plane with Large Energy

TL;DR

This paper investigates the stability of equivariant wave maps from hyperbolic space to the hyperbolic plane, providing evidence for the soliton resolution conjecture for large energy perturbations.

Contribution

It verifies the soliton resolution conjecture for a nonperturbative subset of harmonic maps in the context of equivariant wave maps.

Findings

Confirmed asymptotic stability for a subset of harmonic maps.

Supported the soliton resolution conjecture for large energy wave maps.

Extended previous stability results to nonperturbative regimes.

Abstract

In this paper we continue the analysis of equivariant wave maps from 2-dimensional hyperbolic space into surfaces of revolution that was initiated in [13, 14]. When the target is the hyperbolic plane we proved in [13] the existence and asymptotic stability of a 1-parameter family of finite energy harmonic maps indexed by how far each map wraps around the target. Here we conjecture that each of these harmonic maps is globally asymptotically stable, meaning that the evolution of any arbitrarily large finite energy perturbation of a harmonic map asymptotically resolves into the harmonic map itself plus free radiation. Since such initial data exhaust the energy space, this is the soliton resolution conjecture for this equation. The main result is a verification of this conjecture for a nonperturbative subset of the harmonic maps