A complete classification of finite Morse index solutions to elliptic sine-Gordon equation in the plane

TL;DR

This paper classifies all finite Morse index solutions to the elliptic sine-Gordon equation in the plane, showing they are exactly the known explicit multi-end solutions and analyzing their stability properties.

Contribution

It provides a complete classification of finite Morse index solutions and establishes their non-degeneracy and Morse index in relation to the number of ends.

Findings

All finite Morse index solutions are from the explicit family of multi-end solutions.

Solutions with 2n ends have Morse index n(n-1)/2.

Finite Morse index solutions are non-degenerate with no nontrivial bounded kernel.

Abstract

The elliptic sine-Gordon equation in the plane has a family of explicit multiple-end solutions (soliton-like solutions). We show that all the finite Morse index solutions belong to this family. We also prove they are non-degenerate in the sense that the corresponding linearized operators have no nontrivial bounded kernel. We then show that solutions with $2n$ ends have Morse index $n(n-1)/2.$