In this paper, we use a variational approach to study traveling wave solutions of a gradient system in an infinite strip. As the even-symmetric potential of the system has three local minima, we prove the existence of a traveling wave that propagates from one phase to the other two phases, where these phases corresponds to the three local minima of the potential. To control the asymptotic behavior of the wave at minus infinity, we successfully find a certain convexity condition on the potential, which guarantees the convergence of the wave to a constant state but not to a one-dimensional homoclinic solution or other equilibria. In addition, a non-trivial steady state in $ \mathbb R^2 $ is established by taking a limit of the traveling wave solutions in the strip as the width of the strip tends to infinity.

中文翻译：

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梯度系统三相行波的变分方法

在本文中，我们使用变分方法来研究无限带中梯度系统的行波解。由于系统的偶对称势具有三个局部极小值，我们证明了从一相传播到其他两相的行波的存在，其中这些相对应于势的三个局部极小值。为了控制负无穷处波的渐近行为，我们成功地在势能上找到了一定的凸性条件，这保证了波收敛到恒定状态而不是一维同宿解或其他平衡。此外，当条带的宽度趋于无穷大时，通过取条带中行波解的极限来建立 $\mathbb R^2 $ 中的非平凡稳态。