Heteroclinic traveling waves of 2D parabolic Allen-Cahn systems

Abstract

n this paper we show the existence of traveling waves $w:[0,+\infty )\times {\mathbb{R}}^2\to {\mathbb{R}}^k$ ($k\ge 2$) for the parabolic Allen-Cahn system \begin{equation} \partial_t w - Δw = -\nabla_u V(w) \mbox{ in } [0,+\infty) \times \mathbb{R}^2, \end{equation} satisfying some \textit{heteroclinic} conditions at infinity. The potential $V$ is a non-negative and smooth multi-well potential, which means that its null set is finite and contains at least two elements. The traveling wave $w$ propagates along the horizontal axis according to a speed $c^{\star }>0$ and a profile $\mathfrak{U}$. The profile $\mathfrak{U}$ joins as $x_1\to \pm \infty$ (in a suitable sense) two locally minimizing 1D heteroclinics which have different energies and the speed $c^{\star }$ satisfies certain uniqueness properties. The proof of variational and, in particular, it requires the assumption of an upper bound, depending on $V$, on the difference between the energies of the 1D heteroclinics.

There were some mistakes in the statements and proofs regarding the behavior at infinity of the solutions. These mistakes have been fixed in this version. Other modifications, of less importance, have been made

References in corpus (2)

• Orbital stability of the black soliton to the Gross-Pitaevskii equation
• Asymptotics for two-dimensional vectorial Allen-Cahn systems