Doubling construction for O(m)×O(n) invariant solutions to the Allen–Cahn equation

We construct new families of two-ended O(m)×O(n)-invariant solutions to the Allen–Cahn equation Δu+u−u3=0 in RN+1, with N≥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)×O(n)-invariant manifold, which is asymptotic to the Lawson cone at infinity.