Stationary layered solutions in ${\Bbb R}^2$ for an Allen-Cahn system with multiple well potential

Abstract

We study entire solutions on \({\Bbb R}^2\) of the elliptic system \(-\Delta U + \nabla W(u)=0\) where \(W:{\Bbb R}^2\to{\Bbb R}^2\) is a multiple-well potential. We seek solutions \(U(x_1,x_2)\) which are “heteroclinic,” in two senses: for each fixed \(x_2\in{\Bbb R}\) they connect (at \(x_1=\pm\infty\)) a pair of constant global minima of \(W\), and they connect a pair of distinct one dimensional stationary wave solutions when \(x_2\to\pm \infty\). These solutions describe the local structure of solutions to a reaction-diffusion system near a smooth phase boundary curve. The existence of these heteroclinic solutions demonstrates an unexpected difference between the scalar and vector valued Allen–Cahn equations, namely that in the vectorial case the transition profiles may vary tangentially along the interface. We also consider entire stationary solutions with a “saddle” geometry, which describe the structure of solutions near a crossing point of smooth interfaces.