We establish the existence and robustness of layered, time-periodic solutions to a reaction-diffusion equation in a bounded domain in R n , when the diffusion coefficient is sufficiently small and the reaction term is periodic in time and bistable in the state variable. Our results suggest that these patterned, oscillatory solutions are stable and locally… (More)
We present a theory that enables us to construct heteroclinic connections in closed form for 2uxx = Wu(u), where x ∈ R, u(x) ∈ R 2 and W is a smooth potential with multiple global minima. In particular, multiple connections between global minima are constructed for a class of potentials. With these potentials, numerical simulations for the vector Allen-Cahn… (More)
We rewrite the system ∆u − Wu(u) = 0, for u : R n → R n , in the form div T = 0, where T is an appropriate stress-energy tensor, and derive certain a priori consequences on the solutions. In particular, we point out some differences between two paradigms: the phase-transition system, with target a finite set of points, and the Ginzburg–Landau system, with… (More)
In the present paper we consider the system ∆u−W u (u) = 0, where u : R n → R n , for a class of potentials W : R n → R that possess several global minima and are invariant under a general finite reflection group G. We establish existence of nontrivial entire solutions connecting the global minima of W along certain directions at infinity.
We consider the system where W u (u) = W u1 (u), W u2 (u) , in an equivariant class of functions. We prove that there exists u, a two-dimensional solution, which satisfies the conditions u(x 1 , x 2) → a ± , as x 1 → ±∞, where a + , a − ∈ R 2 are the two global minima of the potential W. We also consider the problem on bounded rectangular domains with… (More)
Mathematically, the problem considered here is that of heteroclinic connections for a system of two second order differential equations of gradient type, in which a small parameter ǫ conveys a singular perturbation. The motivation comes from a multi-order-parameter phase field model developed by Braun et al  and  for the description of crystalline… (More)
We clarify a point in , concerning the derivation of the Plateau angle conditions for the vector-valued Allen-Cahn equation. The Plateau angle conditions for the vector-valued Allen-Cahn equation have recently been rigorously derived in the interesting preprint . However, it is not clear to the author how the estimates (30) and (31) in  are used… (More)
We investigate a model of anisotropic diffuse interfaces in ordered FCC crystals introduced recently by Braun et al and Tanoglu et al [BCMcFW, T, TBCMcF], focusing on parametric conditions which give extreme anisotropy. For a reduced model, we prove existence and stability of plane wave solutions connecting the disordered FCC state with the ordered Cu 3 Au… (More)