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

 1 Department of Mathematics, Shanghai Normal University, Shanghai 200234, China 2 The school of Mathematical Science, Beijing Normal University, Beijing 100875, China 3 College of Science, University of Shanghai for Science and Technology, Shanghai 200093, China

* Corresponding author: Zhixian Yu

Received  June 2020 Revised  August 2020 Published  October 2020

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 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 \$
.
Citation: Zhixian Yu, Rong Yuan, Shaohua Gan. Novel entire solutions in a nonlocal 2-D discrete periodic media for bistable dynamics. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2020314
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