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

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.