Novel entire solutions in a nonlocal 2-D discrete periodic media for bistable dynamics

This paper is concerned with novel entire solutions originating from three pulsating traveling fronts for nonlocal discrete periodic system (NDPS) on 2-D Lattices

$ \begin{align*} \label{eq1.1} u_{i,j}'(t) = \sum\limits_{k_1\in\mathbb{Z}\backslash \{0\}}\sum\limits_{k_2\in\mathbb{Z}\backslash \{0\} }J(k_1,k_2)\Big[u_{i-k_1,j-k_2}(t)- u_{i,j}(t)\Big]+ f_{i,j}(u_{i,j}(t)).\quad \end{align*} $

More precisely, let $ \varphi_{i,j;k}(i cos\theta +j sin\theta+v_{k}t)\,\,(k = 1,2,3) $ be the pulsating traveling front of NDPS with the wave speed $ v_k $ and connecting two different constant states, then NDPS admits an entire solution $ u_{i,j}(t) $, which satisfies

$ \begin{align*} &\ \lim\limits_{t\rightarrow-\infty}\Big\{ \sum\limits_{1\leq k\leq3}\sup\limits_{ p_{k-1}(t)\leq \xi\leq p_k(t)} |u_{i,j}(t)-\varphi_{i,j;k}(\xi+v_{k}t+\theta_{k})|\Big\} = 0, \end{align*} $

where $ \xi = :i \cos\theta +j \sin\theta $, $ v_1<v_2<v_3 $ and $ \theta_{k}\,(k = 1,2) $ is some constant, $ p_0 = -\infty $, $ p_k(t): = -(v_k+v_{k+1})t/2\,\,(k = 1,2) $ and $ p_3 = +\infty $.