Clustering phase transition layers with boundary intersection for an inhomogeneous Allen-Cahn equation

We consider the nonlinear problem of inhomogeneous Allen-Cahn equation

$ \epsilon^2\Delta u+V(y)\,(1-u^2)\,u = 0\quad \mbox{in}\ \Omega, \qquad \frac {\partial u}{\partial \nu} = 0\quad \mbox{on}\ \partial \Omega, $

where $ \Omega $ is a bounded domain in $ \mathbb R^2 $ with smooth boundary, $ \epsilon $ is a small positive parameter, $ \nu $ denotes the unit outward normal of $ \partial \Omega $, $ V $ is a positive smooth function on $ \bar\Omega $. Let $ \Gamma\subset\Omega $ be a smooth curve dividing $ \Omega $ into two disjoint regions and intersecting orthogonally with $ \partial\Omega $ at exactly two points $ P_1 $ and $ P_2 $. Moreover, by considering $ {\mathbb R}^2 $ as a Riemannian manifold with the metric $ g = V(y)\,({\mathrm d}{y}_1^2+{\mathrm d}{y}_2^2) $, we assume that: the curve $ \Gamma $ is a non-degenerate geodesic in the Riemannian manifold $ ({\mathbb R}^2, g) $, the Ricci curvature of the Riemannian manifold $ ({\mathbb R}^2, g) $ along the normal $ \mathbf{n} $ of $ \Gamma $ is positive at $ \Gamma $, the generalized mean curvature of the submanifold $ \partial\Omega $ in $ ({\mathbb R}^2, g) $ vanishes at $ P_1 $ and $ P_2 $. Then for any given integer $ N\geq 2 $, we construct a solution exhibiting $ N $-phase transition layers near $ \Gamma $ (the zero set of the solution has $ N $ components, which are curves connecting $ \partial\Omega $ and directed along the direction of $ \Gamma $) with mutual distance $ O(\epsilon|\log \epsilon|) $, provided that $ \epsilon $ stays away from a discrete set of values to avoid the resonance of the problem. Asymptotic locations of these layers are governed by a Toda system.