Competing Interactions and Traveling Wave Solutions in Lattice Differential Equations

The existence of traveling front solutions to bistable lattice differential equations in the absence of a comparison principle is studied. The results are in the spirit of those in Bates, Chen, and Chmaj in[1], but are applicable to vector equations and to more general limiting systems. An abstract result on the persistence of traveling wave solutions is obtained and is then applied to lattice differential equations with repelling first and/or second neighbor interactions and to some problems with infinite range interactions.


Introduction
We study the existence of traveling wave solutions for lattice differential equations (LDEs) by means of a perturbation argument and Fredholm theory for mixed type functional differential equations. In particular, we prove persistence of traveling waves for a general class of lattice differential equations with bistable nonlinearity. Consider the following equation, Our primary interest is in competing interactions between first and second nearest neighbors when d 1 < 0 and d 2 < 0. We develop a general technique for continuation of solutions of vector dissipative lattice differential equations and obtain results on existence of traveling front solutions for (1.1) when d 1 < 0, 0 < −d 2 ≪ 1 and when d 2 < 0 and |d 1 | ≪ 1.
Our contribution is to develop techniques based upon the implicit function theorem that are applicable for vector equations that are similar to that developed by Bates, Chen, and Chmaj [1] for scalar equations. Whereas in [1] the limiting system is the traveling wave equation associated with the PDE u t = u xx − f (u), we consider, through the use of the Fredholm theory for mixed type functional differential equations [14], limiting equations that may correspond to lattice differential equations. Among the chief motivations in this work (and in [1]) for the use of implicit function theorem based techniques is the desire to handle cases in which there does not exist a comparison principle.
Traveling wave solutions to (1.1) have been extensively studied when d 1 > 0 and d 2 = 0. In particular, the work of Weinberger based upon the development of an abstract comparison principle is applicable to both PDEs and LDEs, although primarily for monostable as opposed to bistable problems. Zinner proved existence of traveling fronts using topological fixed point results [21] and stability [20] in the bistable case. A general stability theory was developed by Chow, Mallet-Paret, and Shen [5] and Shen employed comparison principle techniques to prove results on existence, uniqueness, and stability of traveling fronts in which f ≡ f (u, t) may depend periodically on t. More recently Chen, Guo, and Wu developed a framework for existence, uniqueness, and stability of bistable equations in periodic media [4] and Hupkes and Sandstede [10] prove the existence of traveling pulse solutions for discrete in space Fitz-Hugh Nagumo equations that occur when coupling a relaxation variable to the discrete Nagumo equation ((1.1) with d 1 > 0 and d 2 = 0). Associated with traveling waves for (1.1) when d 1 > 0 and d 2 = 0 is the mixed type functional differential equations −cϕ ′ (ξ) = d 1 (ϕ(ξ − 1) − 2ϕ(ξ) + ϕ(ξ + 1)) − f (ϕ(ξ)) which results from the traveling wave ansatz u j (t) = ϕ(j − ct). Among the important contributions to the study of these types of equations is the pioneering work of Rustichini [17,18], the development by Mallet-Paret of a Fredholm theory for linear mixed type function differential equations [14] and its use to understand the global structure of traveling wave solutions [15]. Exponential dichotomies for these equations were investigated in [6] and [13] and center manifold theory and Lin's method were developed in [8] and [9], respectively.
The case in which d 1 < 0 and d 2 = 0 was investigated in [19] and [2]. In [19] a model was developed for the dynamics of twinned microstructures that arise in martensitic phase transformation, e.g., in shape memory alloys, which led to (1.1) in an overdamped limit. Subsequently, the bistable nonlinearity f (u) = u − H(u − a), H the Heaviside step function, was employed and transform techniques were utilized to determine waveforms and wavespeeds. In [2] the cubic nonlinearity was employed and the problem was converted to a periodic media problem so that the results of [4] could be applied. A wealth of traveling wave solutions of both bistable and monostable type were revealed. Similar techniques may be used to determine traveling fronts when d 2 < 0 and d 1 = 0 which results in two decoupled systems of equations. In [19,2] one of the essential ideas (see also [3]) was to convert to a system in terms of odd and even lattice sites. This effectively allows us to consider connecting orbit problems between vector equilibria as opposed to connecting orbit problems between time independent spatially periodic solutions. Existence and structure of traveling fronts for higher space dimension versions of (1.1) was recently investigated in [11] using comparison principle and continuation techniques. This paper is organized as follows. In section 2 we present some of the notation we will employ and background on Fredholm theory from [14] for linear mixed type functional differential equations. In addition, we summarize two approaches to the existence of traveling wave solutions in lattice differential equations. The first due to Chen, Guo, and Wu [4] provides existence, uniqueness and stability results for traveling front solutions of (1.1) when d 1 and d 2 are positive. The second is due to Bates, Chen, and Chmaj [1] and provides existence of traveling front solutions when d 1 + 4d 2 > 0. Section 3 contains our main results and establishes the persistence of traveling wave solutions for vector equations. In particular, we consider systems of lattice equations and allow, under certain non-restrictive conditions, general limiting systems. In section 4 we consider the application of general results in section 3 to the existence of traveling fronts to (1.1) for values of d 1 , d 2 which even after rewriting as a system (equivalently in a periodic media) do not possess a comparison principle. We end up with conclusions in section 5.

Fredholm Alternative for Lattice Differential Equations
If X, Y are Banach spaces with norms · X , · Y respectively, then we let L(X, Y ) denote the Banach space of bounded linear operators T : X → Y . Denote the kernel and range of T ∈ L(X, Y ) by Recall that T is a Fredholm operator if T satisfies the following: In [14], Mallet-Paret investigated the Fredholm alternative for the following functional differential equations of mixed type, for 1 ≤ p ≤ ∞, where I is some bounded interval, r 1 = 0, and r j = r k , 1 ≤ j < k ≤ N 1 , N 1 ≥ 2. We may write it as 2) and we have the homogeneous equation is a constant matrix, which is independent of x, we denote it by A j,0 and then we may write equation (2.5) We recall Theorem A in Mallet-Paret's paper [14]: [14]) For each p with 1 ≤ p ≤ ∞, Λ L is a Fredholm operator from W 1,p to L p provided that equation −cu ′ (x) = Lu is asymptotically hyperbolic.
We note here that for linear mixed type functional differential equations the standard formula for computation of the Fredholm index is generally not valid, but this is remedied using the spectral flow formula (see [14] Theorem C).

Traveling waves for Bistable Dynamics
In this subsection, we will state the results of the study of the traveling waves of lattice equations for bistable dynamics in [4] and [1]. In [4], consider a general system of spatially discrete reaction diffusion equations for u(t) = {u n (t)} n∈Z : u n (t) = k a n,k u n+k (t) + f n (u n (t)), n ∈ Z, t > 0, (2.6) where the coefficients a n,k are real numbers and have the following assumptions: A1. Periodic medium. There exists a positive integer N such that a n+N,k = a n,k and f n+N (·) = f n (·) ∈ C 2 (R) for all n, k ∈ Z. A2. Existence of ordered, periodic equilibria.
After an appropriate changeof-variables, the equilibria take the form φ − = 0 and φ + = 1. A3. Finite-range interaction. There exists a positive integer k 0 such that a n,k = 0 for |k| > k 0 and for all n ∈ Z. a is,i s+1 −is > 0.
As ǫ → 0, we have It is well-known that (2.8) has a unique traveling wave solution denoted by (c 0 , φ 0 ).
Then there exists a positive constant ǫ * such that for every ǫ ∈ (0, ǫ * ), the problem (2.7) admits a solution (c ǫ , φ ǫ ) satisfying lim Next, with the Fredholm theory in [11] and ideas in [1], we will study the existence of traveling waves to vector LDEs in an abstract framework.

Persistence of Traveling Waves to Lattice Differential Equations with Perturbations
In this section, our goal is to study the persistence of traveling waves of the lattice differential equations, where Λ is defined as in (2.5), that is, A j (x)u(x + r j ) with r 1 = 0, and r j = r k , 1 ≤ j < k ≤ N 1 , N 1 ≥ 2. The perturbed system of (3.1) is of the form, where ǫ > 0 and Bu := We now give the assumptions for the systems of (3.1) and (3.2). We make the following assumption for the nonlinear term: We remark that even though our application examples in next section focus on bistable nonlinearity, (H1) is a more general assumption. Assume that (H2)There exists a traveling wave solution connecting 0 and 1 for (3.1).
We let (c 0 , φ 0 ) be a traveling wave with speed c 0 > 0 for (3.1). We make the following assumption for the perturbed term: For simplicity, let Λ ǫ = Λ + ǫB. We may write (3.2) in It is natural to hope that at least for small ǫ, (3.3) also has a traveling wave solution.
0 are asymptotically hyperbolic then (H4) is satisfied. This is equivalent to check the assumption that L ± 0 are hyperbolic at ±∞. Note that φ 0 (∞) = 1 Then (H4) is equivalent to the following: In applications, we may use (Ĥ4) instead of (H4) if needed since (Ĥ4) can be more easily verified.
To prove Theorem 3.1, with the arguments of perturbation of Fredholm operators, we borrow ideas from [1], which are applicable to vector LDEs. We made assumption (H2) for (3.1) instead of giving some specific equation having a traveling wave solution like (2.8). Existing literature like Theorem 2.2 in Section 2 that Chen, Guo and Wu proved in [4] can provide nice candidates for (3.1) satisfying (H2). To verify (H4), the Fredholm alternative theory (See [14] and [11]) plays an important role. Let X := H 1 (R, R N ). Since L ± 0 are Fredholm operators and dim(K(L ± 0 )) is finite, X can be decomposed by Following the ideas as in [1], we let φ = φ 0 + ψ for ψ ∈ X η and formulate the problem as where (3.7) In some places, we need study the operator of L + ǫ = c 0 ψ ′ − Λ ǫ ψ + γ(φ 0 )ψ, and its adjoint Let (2) There exists a positive constant C 0 , which depends only on F, such that Proof. (1) L + 0 ψ + 0 = 0 follows by differentiating the equation and a direct computation. By Theorem 2.1, dim(K(L + 0 )) = dim(K(L − 0 )), and then there exists Without loss of generality, we assume φ n H 1 = 1. Thus, we have . On the other hand, by the construction, u is in the orthogonal complement of Let c(ψ) be the unique constant such that R(c, ψ)⊥ψ − 0 . Thus we have Proof. It can be verified by direct computation.
Next we prove the case with ǫ = ǫ * . We simply put φ( exists a subsequence u ǫ k that converges uniformly on bounded set. Recall that Let c ǫ * = lim ǫ k →ǫ * c ǫ . This completes the proof. Remark 3.1. Replacing (c 0 , φ 0 ) by (c ǫ * , φ ǫ * ) and following the arguments of the proofs in Theorems 3.1, ǫ * can be extended further unless (H4) is not satisfied.
In [14], Mallet-Paret provided some sufficient conditions for the one dimensional kernel to scalar LDEs and in [11], Hupkes and the first author of current paper generated the results in [14] to vector LDEs, where γ ≥ 0 and ρ ∈ V ⊂ R. Assume that, (HA) A is irreducible (i.e,it is not similar to a block upper-triangular matrix) and nonnegative.

Applications:Existence of Traveling Waves for Mixed Type LDEs
We will introduce four examples in this section. In the first three subsections, we consider equation (1.1). In the last subsection, we consider the perturbations of equation (2.6) with infinity range interactions. Let d = d 1 + 4d 2 . u j is called a stationary solution of (1.1) if u j satisfies d 1 (u j+1 − 2u j + u j−1 ) + d 2 (u j+2 − 2u j + u j−2 ) − f a (u j ) = 0. u j is called a N-Periodic stationary solution of (1.1) if u j is a stationary solution and u j+N = u j .

Traveling Waves Connecting 0 and 1
Define Consider the following equation, For d > 0, it is well-known that the equation (4.2) has a unique traveling wave φ 0 and the speed c 0 . Now we consider the system (4.1). By changing variables, we can make f satisfy (i) in Theorem 2.3. Equation   If (A1) is not satisfied, for d < 0, we will transform our model to a new one which is in the framework of perturbation method developed in the previous section. In section 4.2, we will consider the case with d < 0 but d 1 dominates d 2 in the sense |d 1 | ≫ |d 2 |. In section 4.3, we will deal with the case with d < 0 but d 2 dominates d 1 in the sense |d 2 | ≫ |d 1 |.

Traveling Waves Connecting Two 2-Periodic States
As in the work of Brucal -Hallare and Van Vleck [2], we will use a 2-D transformation. First we write the even and odd nodes of the above equation as x = {x j } j∈Z N and y = {y j } j∈Z N , respectively, and obtain To compute the equilibria, define (x ± , y ± ) by The equilibria satisfy E :

Then substituting into (4.3) we obtain
where d e = d 1 By choosing proper x,y such that d e , d 0 > 0. If d 2 = 0, this is the case studied in [2]. We remark that the case with d 2 = 0 can be easily extended to the case with d 2 ≥ 0.
Proof. The proof follows from the direct computation.
We can pick those equilibria (w ± , x ± , y ± , z ± ) such that after the transformation to 0 and 1, any other 4-periodic state φ = {φ n } n∈Z with φ n ∈ (0, 1), if it exists, is unstable. In this paper, we focus on the cases having bistable dynamics after the transformation. By Theorem 2.2, there exists a traveling wave solution (c 0 , φ 0 ) for (4.12).

Remark 4.2.
In this section we have consideredÃ j such that the results in [4] on existence of traveling waves for bistable problems give monotone waveforms for the limiting system. This yields, via the results in [11], a one dimensional kernel for the linearization about the reference solution. Alternatively, if perturbations include all terms multiplying d 1 , then the limiting system is decoupled and (A4) is not satisfied. However, by considering the even and odd systems independently, the linearization about the reference solution has a two dimensional kernel and the behavior of solutions under perturbation may be analyzed using the bifurcation equations obtained through the Lyapunov-Schmidt reduction.

Traveling Waves for LDEs with Infinite-Range Interactions
In this section, we study the a generalized model of [4] by adding some infinite range interactions. Consider the following: u n (t) = k a n,k u n+k (t) + f n (u n (t)), n ∈ Z, t > 0, (4.14) where the coefficients a n,k are real numbers satisfying k a n,k e kλ < ∞ for any λ ∈ R and satisfy the assumptions (A1,A2,A4,A5). Compared with the equation in [4], the essential difference is in (A3) and (A5), where we remove the assumption (A3), finite range interactions, and consider an infinite sum in (A5).
Let (B k 0 φ) i := |k|<k 0 a n,k e kµ φ i+k for given µ ∈ R. Consider the eigenvalue problem: Lemma 4.4. For each k 0 , if B k 0 is irreducible and quasipositive(i.e, off-diagonal elements are nonnegative), then principal eigenvalue exists, denoted by λ(k 0 ). Moreover, if both λ(k 0 ) and λ(∞) exist, lim Proof. The existence of a principal eigenvalue is followed by Krein-Rutman theorem. Moreover, we have that λ(k 0 ) = lim n→∞ B n k 0 1/n , which implies that lim Proof. This can be proved by modifying the arguments (replacing k 0 , that defines the finite range of interactions, with n, the period of the media) in the proof of Theorem 2 of [4].
Then we have the following theorem.
Proof. The existence of traveling waves follows from the arguments in Section 3. Next we show that monotonicity persists under small perturbations. By the arguments in Theorem 2 of [4] (see Lemma 4.5), a traveling wave must have exponential tails: e (i−ct)λ 1 φ 1 i = λ 1 h + . Note that λ 0 > 0 and λ 1 < 0. We have that ∂u i (t) ∂t has the same sign as |i − ct| > M for some large M. Thus the traveling wave will preserve the monotonicity at the two far ends for small perturbation because the principal eigenvalue will preserve the sign for small perturbation. Obviously, ∂u i (t) ∂t will preserve the sign on i − ct ∈ [−M, M] for small perturbation. This completes the proof.

Conclusion
In this paper we develop an existence theory via perturbation arguments for traveling wave solutions of vector lattice differential equations. Motivation comes from problems in which there is not a comparison principle. In particular, we consider lattice differential equations in which there are repelling first and/or second nearest neighbor interactions. The structure of the kernel (see Proposition 8.2 in [11]) of the linearized operator of the limiting system is central to our analysis. Our general result is modeled after the perturbation arguments in [1]. A possible alternative approach is the Newton/Lyapunov-Schmidt method developed in [7,11]. Finally, we employ the technique developed here to show the existence of traveling waves for bistable lattice differential equations in periodic media with infinite range interactions. Although the results obtained here are primarily of a local nature, they may be extended to global continuation results in certain cases. This necessitates a Fredholm theory for linearized operators that do not satisfy a strict ellipticity conditions such as (A5), e.g., see [1], together with results on the dimension and structure of the kernel. While the Fredholm theory for problems with infinite range interactions is not well developed, the results in [12] apply to certain infinite range interactions.