Interface foliation for an inhomogeneous Allen-Cahn equation in Riemannian manifolds

Let (M, g̃) be an N-dimensional smooth compact Riemannian manifold. We consider the problem ε² Δ _g̃ũ + V(Z̃)ũ(1-ũ²) = 0 in M, where ε > 0 is a small parameter and V is a positive, smooth function in M. Let κ ⊂ M be an (N-1)-dimensional smooth submanifold that divides M into two disjoint components M_±. We assume κ is stationary and non-degenerate relative to the weighted area functional ∫_κ V^1/2. For each integer m ≥ 2, we prove the existence of a sequence ε = ε _ℓ → 0, and two opposite directional solutions with m-transition layers near κ, whose mutual distance is O(ε{pipe}log ε{pipe}). Moreover, the interaction between neighboring layers is governed by a type of Jacobi-Toda system.