Saddle Solutions to Allen-Cahn Equations in Doubly Periodic Media

We consider a class of periodic Allen-Cahn equations (1) - Delta u(x,y) + a(x,y)W' (u(x, y)) = 0, (x,y) is an element of R-2, where a is an element of C(R-2) is an even, periodic, positive function representing a doubly periodic media, and W : R -> R. is a classical double well potential such as the Ginzburg-Landau potential W(s) = (s(2) - 1(2))(2). We show the existence and asymptotic behavior of a saddle solution on the entire plane, which has odd symmetry with respect to both axes, and even symmetry with respect to the line x = y. This result generalizes the classic result on saddle solutions of Allen-Cahn equation in a homogeneous medium.