Periodic motions for multi-wells potentials and layers dynamic for the vector Allen-Cahn equation

TL;DR

This paper investigates periodic orbits in multi-well potentials for the vector Allen-Cahn equation, establishing existence conditions and analyzing layered solutions with slow dynamics.

Contribution

It introduces new conditions for the existence of periodic orbits visiting multiple potential wells and studies the slow motion of layered solutions in the vector Allen-Cahn equation.

Findings

Existence of periodic orbits visiting prescribed neighborhoods of zeros of W.

Derivation of a system of ODEs describing layer dynamics.

Layered solutions exhibit extremely slow motion.

Abstract

We consider a nonnegative potential $W$ that vanishes on a finite set and study the existence of periodic orbits of the equation \[\ddot{u}=W_u(u),\;\;t\in\R,\] that have the property of visiting neighborhoods of zeros of $W$ in a given finite sequence. We give conditions for the existence of such orbits. After introducing the new variable $x=\epsilon t$, $\epsilon>0$ small, these orbits correspond to stationary solutions of the parabolic equation \[u_t=u_{xx}-W_u(u),\;\;x\in(0,1),\;t>0,\] with periodic boundary conditions. In the second paper of the paper we study solutions of this equation that, as the stationary solutions, have a layered structure. We derive a system of ODE that describes the dynamics of the layers and show that their motion is extremely slow.