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

Abstract

This paper is concerned with novel entire solutions originating from three pulsating traveling fronts for nonlocal discrete periodic system (NDPS) on 2-D Lattices u i,j (t) = k 1 ∈Z\{0} k 2 ∈Z\{0} J(k 1 , k 2) u i−k 1 ,j−k 2 (t) − u i,j (t) + f i,j (u i,j (t)). More precisely, let ϕ i,j;k (icosθ + jsinθ + 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 lim t→−∞ 1≤k≤3 sup p k−1 (t)≤ξ≤p k (t) |u i,j (t) − ϕ i,j;k (ξ + v k t + θ k)| = 0, where ξ =: i cos θ + j sin θ, v 1 < v 2 < v 3 and θ k (k = 1, 2) is some constant, p 0 = −∞, p k (t) := −(v k + v k+1)t/2 (k = 1, 2) and p 3 = +∞.