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 …