Uniqueness of Traveling Waves for a Two-Dimensional Bistable Periodic Lattice Dynamical System

We study traveling waves for a two-dimensional lattice dynamical system with bistable nonlinearity in periodic media. The existence and the monotonicity in time of traveling waves can be derived in the same way as the one-dimensional lattice case. In this paper, we derive the uniqueness of nonzero speed traveling waves by using the comparison principle and the sliding method.


Introduction
In this paper, we study the following two-dimensional 2D lattice dynamical system: where f is a C 2 function in R and D 2 u i,j : p i 1,j u i 1,j p i,j u i−1,j q i,j 1 u i,j 1 q i,j u i,j−1 − d i,j u i,j , d i,j : p i 1,j p i,j q i,j 1 q i,j . 1. 2 for some constant a ∈ 0, 1 . For simplicity, we only consider the case when a ∈ 0, 1/2 . We are interested in planar traveling wave solutions of 1.1 such that for any t ∈ R in the direction r, s : cos θ, sin θ for some θ ∈ 0, π/2 . The study of lattice dynamical systems has attracted a lot of attention for past years. In particular, traveling wave solutions are important due to the wide applications of these special solutions. For example, the invading of one species to another can be described by traveling wave solutions see, e.g., 1, 2 . The lattice dynamical system arises, for example, when the habitat is divided into discrete niches in certain biology models. We refer the reader to, for example, 3-9 for monostable nonlinearity and 10-16 for bistable nonlinearity in a one-dimensional lattice. In particular, in 16 the authors studied a very general model with bistable nonlinearity in a 1D lattice. Our purpose of this paper is to extend the result of 16 to the case of multidimensional lattice. For the study of multidimensional lattice, we refer to [17][18][19] . For the simplicity of presentation, we will only consider the 2D lattice dynamical system 1.1 . Our results can be easily extended to the more general case with a convection term or spatially dependent nonlinearity as in 16 .
In a similar manner as that in 16 for 1D lattice case, we can prove the existence of traveling wave solutions of 1.1 -1.6 with profile {U i,j } i,j∈Z and speed c ∈ R, by transforming the problem 1.1 -1.6 into an integral formulation. Moreover, if the speed c > 0, then we can obtainU i,j t < 0 for all i, j ∈ Z and t ∈ R. We will not repeat the proof here and focus on the study of the uniqueness of nonzero speed traveling waves. The uniqueness is in the sense that if there exist two traveling waves with nonzero speeds, then these two speeds are the same, and two wave profiles are the same except a translation. Due to that the nonlinearity is independent of spatial variable, our proof of the uniqueness is simpler and more transparent than that in 16 . In fact, motivated by the work of Fife and McLeod 20 , Lemma 3.1 below provides some estimations in terms of a given traveling wave solution for the solution to the initial value problem for 1.1 with certain initial condition. Moreover, with Lemma 3.1, we employ the idea of moving coordinate and a sliding method to complete the proof of uniqueness see Theorem 3.3 .

3
This paper is organized as follows. In Section 2, we give some preliminaries including a comparison principle. Then we prove uniqueness of traveling wave with nonzero speed in Section 3.

Preliminaries
The following lemma can be easily deduced from 1.5 and 1.6 .
We can determine the sign of the speed c when c / 0 as follows. Proof. For K ≥ max{1, N/|c|}, an integration by parts gives

4
Abstract and Applied Analysis Therefore,

2.3
Sending K to ∞, using 1.5 and Lemma 2.1, it follows that Hence, the lemma follows.
As a simple consequence of Lemma 2.2, we have c 0 if f is of balanced type, that is, 1 0 f s ds 0. Notice that we cannot guarantee the speed c is zero or not by using the method developed in 16 . In fact, for the 1D lattice case, the classical work of Keener 21 indicates that the propagation failure i.e., c 0 occurs when the diffusion coefficient is sufficiently small, even when f is of unbalanced type. A similar result for 1D periodic case can be found in 22 . For our model, the problem for the propagation failure is still open.
Then we have the following comparison principle.
Abstract and Applied Analysis 5 then u i,j t ≥ v i,j t for all t > t 0 , ri sj ct ≤ ri 0 sj 0 . Moreover, if there exists some i 1 , j 1 with ri 1 sj 1 ct 0 < ri 0 sj 0 such that u i 1 ,j 1 t 0 > v i 1 ,j 1 t 0 , then u i,j t > v i,j t for all t > t 0 , ri sj ct < ri 0 sj 0 .
Since the proof is quite similar to the one given in 16, Lemma 1 , we safely omit it here.

Uniqueness
In this section, we will study the uniqueness of traveling waves of 1.1 -1.6 . Firstly, applying a method of Fife-McLeod 20 , we can derive the following result.
for some small > 0.
Proof. We will only consider the case when c > 0. In this case, we haveU c i,j t < 0 for all t ∈ R. First, we let

3.4
Clearly, Φ 0, p is continuous. Fixing with there exists μ 1 > 0, such that f 0 −f p /p ≥ 2μ 1 for all 0 < p ≤ a− . Since f s is continuous, we would find Δ 1 > 0, such that f u − f u p /p ≥ μ 1 for all 0 ≤ u ≤ Δ 1 , 0 < p ≤ a − . By the same reasoning, there exist μ 2 > 0 and Choose l satisfying lim sup ri sj → −∞ u i,j 0 < l < a − < a, and let i 0 , j 0 be determined later. We claim that for some i 0 and j 0 . Since lim sup ri sj → −∞ u i,j 0 < l, there exists m, such that Moreover, since lim ri sj → ∞ U i,j 0 1, there exist i 0 and j 0 , such that Combining 3.8 and 3.9 , we have proved the claim 3.7 . Now, we prove N i,j u ≥ 0. If u i,j t U c

3.10
Divide into three cases.
Choosing δ ≥ l β k / βσ, we have N i,j u ≥ 0 in this case.
Then N i,j u ≥ 0 for all cases. Hence, the second inequality of 3.2 follows from a comparison principle. By the same way, we have the first inequality of 3.2 . This proves the lemma.
Note that we have the following different type of super-and sub-solutions which can be verified by a similar way as that of Lemma 3.1.

Lemma 3.2. Suppose that
We now prove the following uniqueness result.

3.16
Setting the moving coordinate we have for any ξ ∈ R, for all I, J,

3.18
Suppose that c / c. We may assume that c < c. Fixing ξ and sending t → ∞, this leads that either U c I,J ξ ≡ 0 or U c I,J ξ ≡ 1, which is a contradiction. Hence, c c. We now suppress the dependence of c, and we obtain 3.19 For ξ 0 : ri 0 sj 0 N/c, we have U I,J ξ ξ 0 ≤ U I,J ξ ≤ U I,J ξ − ξ 0 , ∀ξ ∈ R, ∀I, J.

3.29
This contradicts with the definition of ξ * . Hence, U i,j ξ U i,j ξ ξ * .
Hence, we obtain the uniqueness up to translations of the traveling wave solution with nonzero speed.