Doubling construction for $O(m)\times O(n)$-invariant solutions to the Allen-Cahn equation

TL;DR

This paper constructs new symmetric solutions to the Allen-Cahn equation in high dimensions, with zero level sets diverging logarithmically from a Lawson cone, advancing understanding of invariant solutions in geometric PDEs.

Contribution

It introduces a novel construction method for $O(m) imes O(n)$-invariant solutions with specific asymptotic behavior, expanding the class of known solutions.

Findings

New families of solutions with two ends and specific symmetry.

Zero level sets diverge logarithmically from Lawson cones.

Method based on Jacobi-Toda system analysis.

Abstract

We construct new families of two-ended $O(m)\times O(n)$-invariant solutions to the Allen- Cahn equation \Delta u+u-u3=0 in $\mathbb{R}^{N+1}$, with $N\ge 7$, whose zero level sets diverge logarithmically from the Lawson cone at infinity. The construction is based on a careful study of the Jacobi-Toda system on a given $O(m)\times O(n)$-invariant manifold, which is asymptotic to the Lawson cone at infinity.