k-ended $O(m) times O(n)$ invariant solutions to the Allen-Cahn equation with infinite Morse index

TL;DR

This paper investigates the existence, asymptotic behavior, and stability of specific invariant solutions to the Allen-Cahn equation in high-dimensional spaces, revealing four families with complex nodal structures and infinite Morse index.

Contribution

It introduces four new families of solutions with logarithmic nodal set corrections, extending previous studies on stable and two-ended solutions.

Findings

Four families of solutions with logarithmic nodal set corrections

Solutions have infinite Morse index

Extends understanding of high-dimensional Allen-Cahn solutions

Abstract

In this work we study existence, asymptotic behaviour and stability properties of $O(m) times O(n)$ invariant solutions of the Allen-Cahn equation $Delta u+u(1-u^2)=0$ in $R^m times R^n$ with $m,n ge 2$, $m+n ge 8$. We exhibit four families of solutions whose nodal sets are smooth logarithmic corrections of the Lawson cone and with infinite Morse index. This work complements the study started by Pacard and Wei (about stable solutions) and by Agudelo, Kowalczyk and Rizzi (about 2 ended solutions).