Journal of Dynamics and Differential Equations ( IF 1.4 ) Pub Date : 2021-04-01 , DOI: 10.1007/s10884-021-09949-5 Giorgio Fusco
We consider a nonnegative potential W that vanishes on a finite set and study the existence of periodic orbits of the equation
$$\begin{aligned} \ddot{u}=W_u(u),\;\;t\in {\mathbb {R}}, \end{aligned}$$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
$$\begin{aligned} u_t=u_{xx}-W_u(u),\;\;x\in (0,1),\;t>0, \end{aligned}$$with periodic boundary conditions. In the second part 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.
中文翻译：
向量Allen–Cahn方程的多势势周期运动和动态层
我们考虑在有限集上消失的非负势W并研究方程的周期轨道的存在
$$ \ begin {aligned} \ ddot {u} = W_u（u），\; \; t \ in {\ mathbb {R}}，\ end {aligned} $$具有在给定有限序列中访问W的零的邻域的特性。我们为此类轨道的存在提供了条件。在引入新变量\（x = \ epsilon t \），\（\ epsilon> 0 \）之后，这些轨道对应于抛物线方程的平稳解
$$ \ begin {aligned} u_t = u_ {xx} -W_u（u），\; \; x \ in（0,1），\; t> 0，\ end {aligned} $$具有周期性边界条件。在本文的第二部分中，我们研究了作为固定解具有分层结构的该方程的解。我们派生了一个ODE系统，该系统描述了层的动力学，并表明它们的运动极其缓慢。
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