Periodic solutions of Allen-Cahn system with the fractional Laplacian (Q2202215)

In the present article, the authors deal with the following nonlinear Allen-Cahn system with fractional Laplacian \[ (-\partial_{xx})^s\mathbf{u}(x) + \nabla F(\mathbf{u}(x))=0 \text{ in } \mathbb{R}\tag{1} \] where \(s\in (0,1)\), the function \(F: \mathbb{R}^2\to \mathbb{R}\) is a smooth double-well potential with wells at \(\mathbf{b}_1\) and \(\mathbf{b}_2\), \(\lim_{|\mathbf{w}|\to \infty} F(\mathbf{w}) = \infty\) and \[ \begin{aligned} &F(\mathbf{b}_i) = 0 < F(\mathbf{w}) \forall \mathbf{w} \neq \mathbf{b}_i,\;\;i=1,2,\\ &\nabla F(\mathbf{w})\cdot \mathbf{w} \geq 0 \text{ for } |\mathbf{w}|\geq 1. \end{aligned}\tag{2} \] The fractional Laplacian operator \((-\partial_{xx})^s\) is defined in the Cafarelli-Silvestre's sense, indeed, problem (1) can be view as \[ \begin{aligned} &\operatorname{div}(y^a\nabla \mathbf{U}) = 0 \text{ in } \mathbb{R}_{+}^{2} \\ & \frac{\partial \mathbf{U}}{\partial \nu^a} = -\nabla F(\mathbf{U}) \text{ on } \mathbb{R}. \end{aligned}\tag{3} \] If \(F(u,v)\) is even with respect to \(u\) and \(v\) respectively, by using variational methods the authors show the existence of \(T_1>0\) such that for any \(T>T_1\), the systems (1) admits a periodic solution \(\mathbf{u}_T\) with period \(T\) and \[ |\mathbf{u}_T| \leq 1. \] Furthermore, the authors have established a Hamiltonian identity for periodic solutions.