Doubly Periodic Traveling Waves in a Cellular Neural Network with Linear Reaction

Advances in Difference Equations Volume 2009, Article ID 243245,29pages doi:10.1155/2009/243245

Research Article

Doubly Periodic Traveling Waves in a Cellular Neural Network with Linear Reaction

Jian Jhong Lin and Sui Sun Cheng

Department of Mathematics, Tsing Hua University, Hsinchu 30043, Taiwan

Correspondence should be addressed to Sui Sun Cheng,sscheng@math.nthu.edu.tw Received 4 June 2009; Accepted 13 October 2009

Recommended by Roderick Melnik

Szekeley observed that the dynamic pattern of the locomotion of salamanders can be explained by periodic vector sequences generated by logical neural networks. Such sequences can mathematically be described by “doubly periodic traveling waves” and therefore it is of interest to propose dynamic models that may produce such waves. One such dynamic network model is built here based on reaction-diffusion principles and a complete discussion is given for the existence of doubly periodic waves as outputs. Since there are 2 parameters in our model and 4 a priori unknown parameters involved in our search of solutions, our results are nontrivial. The reaction term in our model is a linear function and hence our results can also be interpreted as existence criteria for solutions of a nontrivial linear problem depending on 6 parameters.

Copyrightq2009 J. J. Lin and S. S. Cheng. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1. Introduction

Szekely in 1 studied the locomotion of salamanders and showed that a bipolar neural network may generate dynamic rhythms that mimic the “sequential” contraction and relaxation of four muscle pools that govern the movements of these animals. What is interesting is that we may explain the correct sequential rhythm by means of the transition of state values of four differentartificialneurons and the sequential rhythm can be explained in terms of an 8-periodic vector sequence and subsequently in terms of a “doubly periodic traveling wave solution” of the dynamic bipolar cellular neural network.

Similar dynamiclocomotivepatterns can be observed in many animal behaviors and therefore we need not repeat the same description in1. Instead, we may use “simplified”

snorkeling or walking patterns to motivate our study here. When snorkeling, we need to float on water with our faces downward, stretch out our arms forward, and expand our legs backward. Then our legs must move alternatively. More precisely, one leg kicks downward and another moves upward alternatively.

Letv0andv1 be two neuron pools controlling our right and left legs, respectively, so that our leg moves upward if the state value of the corresponding neuron pool is 1,and

−1 0 1 u

0 2 4 6 8 10 12

i

0 5

10 15

20 25

30 35

40

t

Figure 1: Doubly periodic traveling wave.

downward if the state value of the corresponding neuron pool is −1. Let v₀^t and v₁^t be the state values ofv0 andv1 during the time staget,wheret ∈ N {0,1,2, . . .}.Then the movements of our legs in terms ofv^t₀ , v^t₁ , t∈N,will form a 2-periodic sequential pattern

−1,1−→1,−1−→−1,1 → 1,−1−→ · · · 1.1

or

1,−1−→−1,1−→1,−1−→1,1−→ · · ·. 1.2

If we setv^t_i v_i^t_{mod 2} for anyt∈Nandi∈Z{0,±1,±2, . . .},then it is easy to check that

v^t1_i v^t_i1 ∀i∈Z, t∈N

temporal-spatial transition condition , v^t2_i v_i^t ∀i∈Z, t∈N

temporal periodicity condition , v^t_i v^t_i2 ∀i∈Z, t∈N

spatial periodicity condition .

1.3

Such a sequence {v_i^t} may be called a “doubly periodic traveling wave” see Figure 1.

Now we need to face the following important issue as in neuromorphic engineering.

Can we build artificial neural networks which can support dynamic patterns similar to {v^t₀ , v₁^t}_t∈N? Besides this issue, there are other related questions. For example, can we build nonlogical networks that can support different types of graded dynamic patterns remember an animal can walk, run, jump, and so forth, with different strength?

To this end, in 2, we build a nonlogical neural network and showed the exact conditions such doubly periodic traveling wave solutions may or may not be generated by it. The network in2has a linear “diffusion part” and a nonlinear “reaction part.” However,

the reaction part consists of a quadratic polynomial so that the investigation is reduced to a linear and homogeneous problem. It is therefore of great interests to build networks with general polynomials as reaction terms. This job is carried out in two stages. The first stage results in the present paper and we consider linear functions as our reaction functions. In a subsequent paper, as a report of the second stage investigation, we consider polynomials with more general formsee the statement after2.11.

2. The Model

We briefly recall the diffusion-reaction network in 2. In the following, we set N {0,1,2, . . .},Z{. . . ,−2,−1,0,1,2, . . .}andZ{1,2,3, . . .}.For anyx∈R,we also usexto denote the greatest integer part ofx.Suppose thatv₀, . . . , v_Υ−1are neuron pools, whereΥ≥1, placedin a counterclockwise manneron the vertices of a regular polygon such that each neuron poolvihas exactly two neighbors,v_i−1andv_i1,wherei∈ {0, . . . ,Υ−1}.For the sake of convenience, we have setv₀ v₋₁andv₁ v_Υto reflect the fact that these neuron pools are placed on the vertices of a regular polygon. For the same reason, we definev_i v_imodΥ for anyi∈ Zand let eachv^t_i be the state value of theith unitv_i in the time periodt ∈ N.

During the time periodt, if the valuev^t_i of theith unit is higher thanv^t_i−1, we assume that

“information” will flow from theith unit to its neighbor. The subsequent change of the state value of theith unit isv_i^t1−v_i^t, and it is reasonable to postulate that it is proportional to the differencev^t_i −v^t_i−1, say,αv_i^t−v_i−1^t, whereαis a proportionality constant. Similarly, information is assumed to flow from thei1-unit to theith unit if v_i1^t > v^t_i . Thus, it is reasonable that the total effect is

v^t1_i −v^t_i α

v^t_i−1−v^t_i α

v_i1^t −v^t_i α

v_i1^t −2v_i^tv^t_i−1

, i∈Z, t∈N. 2.1

If we now assume further that a control or reaction mechanism is imposed, a slightly more complicated nonhomogeneous model such as the following

v^t1_i −v^t_i α

v^t_i1−v^t_i α

v_i−1^t −v^t_i g

v^t_i

∀i∈Z, t∈N 2.2

may result. In the above model, we assume thatgis a function andα∈R.

The existence and uniqueness ofrealsolutions of2.2is easy to see. Indeed, if the realinitial distribution{v⁰_i }_i∈Zis known, then we may calculate successively the sequence

v₋₁¹, v¹₀ , v₁¹;v¹₋₂, v₋₁², v²₀ , v₁², v¹₂ , . . . 2.3

in a unique manner, which will give rise to a unique solution{v^t_i }_t∈N,i∈Zof2.2. Motivated by our example above, we want to find solutions that satisfy

v^tτ_i v^t_iδ ∀i∈Z, t∈N, 2.4

v^tΔ_i v_i^t ∀i∈Z, t∈N, 2.5

v_i^t v_iΥ^t ∀i∈Z, t∈N, 2.6

whereτ,Δ,Υ∈Zandδ∈Z.It is clear that equations in1.3are special cases of2.4,2.5, and2.6, respectively.

Suppose that v{v^t_i }is a double sequence satisfying2.4for someτ ∈Zandδ∈Z.

Then it is clear that

v^tkτ_i v^t_ikδ for any i∈Z, t∈N, 2.7

wherek ∈Z.Hence when we want to find any solution{v^t_i }of2.2satisfying2.4, it is sufficient to find the solution of2.2satisfying

v^tτ/q_i v^t_iδ/q, 2.8

whereqis the greatest common divisorτ, δofτandδ.For this reason, we will pay attention to the condition thatτ, δ 1.Formally, given anyτ ∈Zandδ ∈Zwithτ, δ 1,a real double sequence{v^t_i }_t∈N,i∈Zis called a traveling wave with velocity−δ/τif

v_i^tτv^t_iδ, t∈N, i∈Z. 2.9

In caseδ0 andτ 1,our traveling wave is also called a standing wave.

Next, recall that a positive integer ω is called a period of a sequence ϕ {ϕm} if ϕ_mωϕmfor allm∈Z. Furthermore, ifω∈Zis the least among all periods of a sequence ϕ,then ϕis said to beω-periodic. It is clear that if a sequenceϕ is periodic, then the least number of all itspositiveperiods exists. It is easy to see the following relation between the least period and a period of a periodic sequence.

Lemma 2.1. If y {yi}isω-periodic andω₁ is a period of y,thenωis a factor ofω₁,orω mod ω10.

We may extend the above concept of periodic sequences to double sequences. Suppose that v {v^t_i }is a real double sequence. Ifξ ∈Z such thatv^t_iξ v_i^t for alliandt,thenξ is called a spatial period of v.Similarly, ifη∈Zsuch thatv^tη_i v_i^tfor alliandt,thenη is called a temporal period of v. Furthermore, ifξis the least among all spatial periods of v, then v is called spatialξ-periodic, and ifηis the least among all temporal periods of v,then v is called temporalη-periodic.

In seeking solutions of2.2that satisfy2.5and2.6, in view ofLemma 2.1, there is no loss of generality to assume that the numbers Δand Υare the least spatial and the

least temporal periods of the sought solution. Therefore, from here onward, we will seek such doubly-periodic traveling wave solutions of2.2. More precisely, given any function g, α ∈R, δ ∈ZandΔ,Υ, τ ∈Z withτ, δ 1,in this paper, we will mainly be concerned with the traveling wave solutions of2.2with velocity−δ/τwhich are also spatialΥ-periodic and temporalΔ-periodic. For convenience, we call such solutionsΔ,Υ-periodic traveling wave solutions of2.2with velocity−δ/τ.

In general, the control functiongin2.2can be selected in many different ways. But naturally, we should start with the trivial polynomial and general polynomials of the form

gx κfx:κx−r1x−r2· · ·x−rn, 2.10

wherer1, r2, . . . , rn are real numbers, andκis a real parameter. In2, the trivial polynomial and the quadratic polynomialfx x² are considered. In this paper, we will consider the linear case, namely,

fx 1 for x∈R or fx x−r forx∈R, wherer∈R, 2.11

while the cases wherer1, r2, . . . , rn are mutually distinct andn ≥ 2 will be considered in a subsequent paperfor the important reason that quite distinct techniques are needed.

Since the trivial polynomial is considered in2, we may avoid the case whereκ0.

A further simplification of2.11is possible in view of the following translation invariance.

Lemma 2.2. Letτ,Δ,Υ∈Z, δ∈Zwithτ, δ 1 andα, κ, r∈Rwithκ /0.Then v{v_i^t}is a Δ,Υ-periodic traveling wave solution with velocity−δ/τfor the following equation:

v^t1_i −v^t_i α

v_i1^t −2v_i^tv^t_i−1 κ

v^t_i −r

, i∈Z, t∈N, 2.12

if, and only if, y{y_i^t}{v_i^t−r}is aΔ,Υ-periodic traveling wave solution with velocity−δ/τ for the following equation

y_i^t1−y^t_i α

y_i1^t −2y^t_i y^t_i−1

κy_i^t, i∈Z, t∈N. 2.13

Therefore, from now on, we assume in2.2that

α∈R, gκf, 2.14

where

κ /0,

fx 1 forx∈R, or, fx x forx∈R. 2.15

As for the traveling wave solutions, we also have the following reflection invariance resulta direct verification is easy and can be found in2.

Lemma 2.3cf. proof of2, Theorem 3. Given anyδ∈Z\ {0}andτ ∈Z withτ, δ 1.If {v^t_i }is a traveling wave solution of 2.2with velocity−δ/τ,then{w_i^t}{v^t_−i}is also a traveling wave solution of 2.2with velocityδ/τ.

Let−δ ∈ Z andΔ,Υ, τ ∈ Z,where τ, δ 1.Suppose that v {v_i^t} is aΔ,Υ- periodic traveling wave solution of2.2with velocity −δ/τ. Then it is easy to check that w {w^t_i } {v^t_−i}is also temporal Δ-periodic and spatialΥ-periodic. From this fact and Lemma 2.3, when we want to consider theΔ,Υ-periodic traveling wave solutions of2.2 with velocity−δ/τ, it is sufficient to consider theΔ,Υ-periodic traveling wave solutions of 2.2with velocityδ/τ. In conclusion, from now on, we may restrict our attention to the case where

τ ∈Z, δ∈N with τ, δ 1. 2.16

3. Basic Facts

Some additional basic facts are needed. Let us state these as follows. First, letA_ξbe a circulant matrix defined by

A2

2 −2

−2 2

,

A_ξ

⎡

⎢⎢

⎢⎢

⎢⎢

⎢⎢

⎢⎢

⎢⎣

2 −1 0 −1

−1 2 −1 0

· · ·

· · ·

0 −1 2 −1

−1 0 −1 2

⎤

⎥⎥

⎥⎥

⎥⎥

⎥⎥

⎥⎥

⎥⎦

ξ×ξ

, ξ≥3.

3.1

Second, we set

λ^i,ξ4sin² iπ

ξ

, i∈Z, ξ∈Z, 3.2

u^i,ξ_m 1

ξcos2miπ/ξ sin2miπ/ξ, m, i∈Z; ξ∈Z. 3.3

It is knownsee, e.g.,3that for anyξ≥2,the eigenvalues ofA_ξ areλ^1,ξ, . . . , λ^ξ,ξand the eigenvector corresponding toλ^i,ξis

u^i,ξ

u^i,ξ₁ , . . . , u^i,ξ_{ξ †}

fori∈ {1, . . . , ξ}, 3.4

and that u^1,ξ, u^2,ξ, . . . , u^ξ,ξ are orthonormal. It is also clear that u^0,ξ u^ξ,ξ, λ^0,ξ λ^ξ,ξ, λ^i,ξλ^ξ−i,ξ, and

u^ξ−i,ξ_m 1

ξcos2miπ/ξ−sin2miπ/ξ ∀m, i∈Z. 3.5

Therefore,λ^0,ξ, . . . , λ^ξ/2,ξare all distinct eigenvalues ofAξwith corresponding eigenspaces span{u^ξ},span{u¹, u^ξ−1}, . . . ,span{u^ξ/2, u^ξ−ξ/2},respectively.

Given any finite sequencev {v1, v2, . . . , vξ}or vectorv

v1, v2, . . . , vξ

_†

, where ξ≥1, itsperiodicextension is the sequencev{v_i}_i∈Zdefined by

vivimodξ, i∈Z. 3.6

Suppose that Υ,Δ ∈ Z and τ, δ satisfy 2.16. When we want to know whether a double sequence is aΔ,Υ-periodic traveling wave solution of2.2with velocity−δ/τ,the following two results will be useful.

Lemma 3.1. Letξ, η∈Zwithξ≥2 and letu^i,ξbe defined by3.4.

iSupposeξ ≥ 4.Letj, k ∈ {1, . . . ,ξ/2}withj /kanda, b, c, d ∈ Rsuch thatau^j,ξ bu^ξ−j,ξandcu^k,ξdu^ξ−k,ξare both nonzero vectors. Thenηis a period of the extension of the vectorau^j,ξbu^ξ−j,ξcu^k,ξdu^ξ−k,ξif and only ifηj/ξ∈Zandηk/ξ∈Z. iiSuppose ξ ≥ 3.Let j ∈ {1, . . . ,ξ/2} and a, b, c ∈ Rsuch that bu^ξ−j,ξcu^j,ξ is a

nonzero vector. Thenau^ξ,ξbu^ξ−j,ξcu^j,ξisξ-periodic if and only ifj, ξ 1.

iiiSupposeξ2.Leta, b∈Rsuch thatb /0.Thenau^2,2bu^1,2is 2-periodic.

Proof. To seei, we need to consider five mutually exclusive and exhaustive cases:aj, k∈ {1, . . . ,ξ/2−1};b ξ is odd, j ∈ {1, . . . ,ξ/2−1} andk ξ−1/2;c ξ is odd, k ∈ {1, . . . ,ξ/2−1}andj ξ−1/2;dξis even,j ∈ {1, . . . ,ξ/2−1}andk ξ/2;eξis even,k∈ {1, . . . ,ξ/2−1}andjξ/2.

Suppose that caseaholds. Take

uau^j,ξbu^ξ−j,ξcu^k,ξdu^ξ−k,ξ, 3.7

wherea, b, c, d ∈Rsuch thatau^j,ξbu^ξ−j,ξ andcu^k,ξdu^ξ−k,ξare both nonzero vectors.

Letu{u_i}_i∈Zbe the extension ofu,so thatu_iu_imodξfori∈Z.Then it is clear that for any i∈Z,

u_iau^j,ξ_i bu^ξ−j,ξ_i cu^k,ξ_i du^ξ−k,ξ_i 1

ξ

abcos2ijπ

ξ a−bsin2ijπ

ξ cdcos2ikπ

ξ c−dsin2ikπ ξ

.

3.8

By direct computation, we also have

u_iη 1

ξcos2ηjπ ξ

abcos2ijπ

ξ a−bsin2ijπ ξ

1

ξsin2ηjπ ξ

a−bcos2ijπ

ξ −absin2ijπ ξ

1

ξcos2ηkπ ξ

cdcos2ikπ

ξ c−dsin2ikπ ξ

1

ξsin2ηkπ ξ

c−dcos2ikπ

ξ −cdsin2ikπ ξ

.

3.9

By3.8and3.9, we see thatηis a period ofu, that is,ui−u_iη 0 for alli∈Z,if, and only if, given anyi∈Z,

0 1 ξ

cos2ηjπ ξ −1

abcos2ijπ

ξ a−bsin2ijπ ξ

1 ξ

cos2ηkπ ξ −1

cdcos2ikπ

ξ c−dsin2ikπ ξ

1

ξsin2ηjπ ξ

a−bcos2ijπ

ξ −absin2ijπ ξ

1

ξsin2ηkπ ξ

c−dcos2ikπ

ξ −cdsin2ikπ ξ

.

3.10

By3.3and3.5, we may rewrite3.10as

0

cos2ηjπ

ξ −1

au^j,ξ_i bu^ξ−j,ξ_i

sin2ηjπ ξ

−bu^j,ξ_i au^ξ−j,ξ_i

cos2ηkπ

ξ −1

cu^k,ξ_i du^ξ−k,ξ_i

sin2ηkπ ξ

−cu^k,ξ_i du^ξ−k,ξ_i .

3.11

By3.3again, we haveu^i,ξ_mξ u^i,ξ_m for eachi, m∈Z.Hence we see thatηis a period ofuif, and only if,

0, . . . ,0^†

cos2ηjπ

ξ −1

au^j,ξbu^ξ−j,ξ

sin2ηjπ ξ

−bu^j,ξau^ξ−j,ξ

×

cos2ηkπ

ξ −1

cu^k,ξdu^ξ−k,ξ

sin2ηkπ ξ

−cu^k,ξdu^ξ−k,ξ .

3.12

Note thatj ∈ {1, . . . ,ξ/2−1}implies thatu^j,ξandu^ξ−j,ξ are distinct and hence they are linearly independent. Thus, the fact thatau^j,ξbu^ξ−j,ξis not a zero vector implies|a||b|/0.

Similarly, we also have|c||d|/0.Then it is easy to check that au^j,ξ bu^ξ−j,ξ,−bu^j,ξ au^ξ−j,ξ, cu^k,ξdu^ξ−k,ξand−cu^k,ξdu^ξ−k,ξare linear independent. Hence we have thatη is a period ofuif and only if

cos2ηjπ

ξ −1cos2ηkπ

ξ −1sin2ηjπ

ξ sin2ηkπ

ξ 0. 3.13

In other words,ηis a period ofuif, and only if,ηj/ξ∈Zandηk/ξ∈Z.

The other casesb–ecan be proved in similar manners and hence their proofs are skipped.

To proveii, we first setuau^ξ,ξbu^ξ−j,ξcu^j,ξ.As ini, we also know thatηis a period ofu{ui}_i∈Z,whereuiuimodξ,if and only ifηj/ξ∈Z.That is,

η∈Z| ηj ξ ∈Z

η∈Z|η is a period ofu

. 3.14

Supposeξ, j 1.Ifηj/ξ∈Zfor someη∈Z,then we haveη0 modξbecauseξ, j 1.

Hence we have

min

η∈Z|η is a period ofu min

η∈Z| ηj ξ ∈Z

ξ. 3.15

In other words, ifξ, j 1,thenu isξ-periodic. Next, supposeξ, j η1/1; that is, there exists someξ1, j1 >1 such thatξη1ξ1andj η1j1.Note thatj < ξand hence we also have η₁ < ξ₁.Sinceξ η₁ξ₁, η₁ < ξandη₁ > 1,we have 1 < ξ₁ < ξ.Takingη ξ₁,then we have ηj/ξ∈Z.Henceηis a period ofuandη < ξ.That is,uis notξ-periodic. In conclusion, ifuis ξ-periodic, thenξ, j 1.

The proof of iii is done by recalling that u^2,2 1/√

21,1^† and u^1,2 1/√

2−1,1^†and checking thatau^2,2bu^1,2is truly 2-periodic. The proof is complete.

The above can be used, as we will see later, to determine the spatial periods of some special double sequences.

Lemma 3.2. Letu^i,ξ be defined by3.4. Letξ ≥ 3, j ∈ {0,1, . . . ,ξ/2}andk ∈ {1, . . . ,ξ/2}

withj /k. Let further

uau^j,ξbu^ξ−j,ξcu^k,ξdu^ξ−k,ξ, u−au^j,ξ−bu^ξ−j,ξcu^k,ξdu^ξ−k,ξ,

3.16

wherea, b, c, d∈Rsuch thatau^j,ξbu^ξ−j,ξis a nonzero vector. Define v{v_i^t}by

v^t_i

i∈Z

⎧⎨

⎩

u, iftis odd,

u, iftis even. 3.17

iSuppose thatj /0 andcu^k,ξdu^ξ−k,ξis a nonzero vector. Then v is spatialξ-periodic if, and only if,ηj/ξ /∈Zorηk/ξ /∈Zfor anyη∈ {1, . . . , ξ−1}withη|ξ.

iiSuppose thatj 0 andcu^k,ξ_i du^ξ−k,ξ_i is a nonzero vector. Then v is spatialξ-periodic if, and only if,k, ξ 1.

iiiSuppose thatcu^k,ξdu^ξ−k,ξis a zero vector. Then v is spatialξ-periodic if, and only if, j, ξ 1.

Proof. To seei, suppose thatj /0 andcu^k,ξdu^ξ−k,ξis a nonzero vector.Note that the fact thatj, k ∈ {1, . . . ,ξ/2}withj /kimpliesξ ≥ 4.ByLemma 3.1i,ηis a period ofuif, and only if,ηj/ξ∈Zandηk/ξ∈Z. ByLemma 3.1iagain,ηj/ξ∈Zandηk/ξ∈Zif, and only if,ηis a period ofu.Hence the least period ofuis the same asuand v is spatialξ-periodic if, and only if,uisξ-periodic. Note thatξis a period ofu. ByLemma 2.1andLemma 3.1i, we haveuisξ-periodic if and only ifηj/ξ /∈Zorηk/ξ /∈Zfor anyη∈ {1, . . . , ξ−1}withη|ξ.

The assertionsiiandiiican be proved in similar manners. The proof is complete.

Lemma 3.3. Letξbe even withξ≥3 and letu^i,ξbe defined by3.4. Letj, k∈ {1, . . . ,ξ/2}and

uau^j,ξbu^ξ−j,ξcu^k,ξdu^ξ−k,ξ, u−au^j,ξ−bu^ξ−j,ξcu^k,ξdu^ξ−k,ξ,

3.18

wherea, b, c, d∈Rsuch thatau^j,ξ_i bu^ξ−j,ξ_i is a nonzero vector. Let v{v^t_i }be defined by

v^t_i

i∈Z

⎧⎨

⎩

u, iftis odd,

u, iftis even. 3.19

iSuppose thatcu^k,ξdu^ξ−k,ξis a nonzero vector. Thenv_i^t1 v^t_iξ/2 for alli∈Zand t∈Nif and only ifjis odd andkis even.

iiSuppose thatcu^k,ξdu^ξ−k,ξis a zero vector. Thenv_i^t1v^t_iξ/2 for alli∈Zandt∈N if and only ifjis odd.

Proof. To seei, we first suppose thatjis odd andkis even. Note that

v_i⁰ 1 ξ

abcos2ijπ

ξ a−bsin2ijπ

ξ cdcos2ikπ

ξ c−dsin2ikπ ξ

, 3.20

v¹_i 1 ξ

−abcos2ijπ

ξ −a−bsin2ijπ

ξ cdcos2ikπ

ξ c−dsin2ikπ ξ

. 3.21

For anys, i∈Z,it is clear that

cos2iξ/2sπ

ξ

⎧⎪

⎪⎪

⎨

⎪⎪

⎪⎩

cos2isπ

ξ ifsis even,

−cos2isπ

ξ ifsis odd,

sin2iξ/2sπ

ξ

⎧⎪

⎪⎪

⎨

⎪⎪

⎪⎩

sin2isπ

ξ ifsis even,

−sin2isπ

ξ ifsis odd.

3.22

Sincejis odd andkis even, by3.22, it is easy to see thatv¹_iv⁰_iξ/2for alli∈Z.By the definition of v,we also have

v^t2_i v_i^t, v^t_i v^t_iξ ∀i∈Z, t∈N. 3.23

In particular, we havev_i^t1v^t_iξ/2for alli∈Zandt∈N.

For the converse, suppose thatj is even orkis odd. We first focus on the case thatj andkare both even. By3.20and3.22, we have

v⁰_iξ/2 1 ξ

abcos2ijπ

ξ a−bsin2ijπ

ξ cdcos2ikπ

ξ c−dsin2ikπ ξ

. 3.24 Ifv^t1_i v^t_iξ/2for alli∈Zandt∈N,it is clear thatv_i¹ v_iξ/2⁰ for alli∈Z. By3.21and 3.24, we have

2abcos2ijπ

ξ 2a−bsin2ijπ

ξ 0 ∀i∈Z. 3.25

That is, 2au^j,ξbu^ξ−j,ξ 0.This is contrary to our assumption. That is, ifj, kare both even, then we havev_i^t1/v^t_iξ/2for somet∈Nandi∈Z.By similar arguments, in case wherej, k are both odd or wherej is even andkis odd, we also havev_i^t1/v^t_iξ/2for somet∈Nand i∈Z.In summary, ifv_i^t1v^t_iξ/2for alli, t,thenjis odd andkis even.

The assertioniiis proved in a manner similar to that ofi. The proof is complete.

4. Necessary Conditions

Let Υ,Δ ∈ Z,in this section, we want to find the necessary and sufficient conditions for Δ,Υ-periodic traveling wave solutions of2.2with velocity−δ/τ,under the assumptions 2.14,2.15, and2.16.

We first consider the case wherefx 1 for allx∈R.Then we may rewrite2.2as v_i^t1−v_i^tα

v^t_i−1−2v^t_i v^t_i1

κ, i∈Z, t∈N, κ /0. 4.1

Suppose that v {v^t_i }is a Δ,Υ-periodic traveling wave solutions of4.1with velocity

−δ/τ. For anyl, s∈Z,it is clear that Δ

t1

Υ i1

v^ts_il ^Δ

t1

Υ i1

v^t_i . 4.2

Then we have

0^Δ

t1

Υ i1

v_i^t1−v^t_i α

Δ t1

Υ i1

v^t_i−1−2v^t_i v^t_i1 ^Δ

t1

Υ i1

κ ΔΥκ /0. 4.3

This is a contradiction. In other words,Δ,Υ-periodic traveling wave solutions of4.1with velocity−δ/τdo not exist.

Next, we consider the casefx xand focus on the equation v^t1_i −v_i^t α

v^t_i−1−2v^t_i v_i1^t

κv_i^t, i∈Z, t∈N, κ /0. 4.4

Before dealing with this case, we give some necessary conditions for the existence ofΔ,Υ- periodic traveling wave solutions of4.4with velocity−δ/τ.

Lemma 4.1. Letα, κ ∈ Rwithκ /0 andτ, δsatisfy2.16. Suppose that v {v_i^t}is aΔ,Υ- periodic traveling wave solution of 4.4with velocity −δ/τ,where Δ 1 and Υ > 1.Then the matrixκI_Υ−αA_Υis not invertible and

v⁰₁ , . . . , v_Υ⁰_†

is a nonzero vector in kerκI_Υ−αA_Υ.

Proof. Let v{v^t_i }be a1,Υ-periodic traveling wave solution of2.2with velocity−δ/τ.

It is clear that

v^t1_i v^t_i , v_i^tv_iΥ^t ∀i, t. 4.5

Since v satisfies4.4, by4.5, we have κIΥ−αA_Υ

v₁^t, . . . , v_Υ^t_†

0, . . . ,0^† ∀t∈N. 4.6

This fact implies that

v^t₁ , . . . , v_Υ^t_†

is a vector in kerκI_Υ−αA_Υ.IfκI_Υ−αA_Υis invertible or

v₁^t, . . . , v^t_{Υ †}

0, by direct computation, we have v^t₁ , . . . , v^t_{Υ †}

0, . . . ,0^† ∀t∈N, 4.7

and hencev^t_i 0 for allt ∈ N andi ∈ Z.This is contrary toΔbeing the least among all spatial periods andΔ>1. That is,κI_Υ−αA_Υis not invertible and

v⁰₁ , . . . , v⁰_{Υ †}

is a nonzero vector in kerκI_Υ−αA_Υ.The proof is complete.

Lemma 4.2. Letα, κ ∈ Rwithκ /0 andτ, δsatisfy2.16. Suppose that v {v_i^t}is aΔ,Υ- periodic traveling wave solution of 4.4with velocity−δ/τ, whereΔ > 1 andΥ 1.ThenΔ 2, κ−2,eachv⁰_i /0 and

v^t_i

⎧⎨

⎩

v⁰_i , iftis even, i∈Z,

−v⁰_i , iftis odd, i∈Z. 4.8

Proof. From the assumption of v,we have

v_i^tΔv^t_i , v^t_i v^t_i1 ∀i, t. 4.9 Note that v also satisfies4.4. Hence by4.9and computation, we have

v^t1_i 1κ^t1v⁰_i , i∈Z, t∈N. 4.10

Ifv_j⁰0 for somej,by4.9and4.10, we havev⁰_i 0 for alliandv^t_i 0 for anyi, t.This is contrary toΔbeing the least among all temporal periods andΔ>1.Hence we havev_i⁰/0 for alli.Then it is clear thatv^t_i is divergent ast → ∞if|1κ|>1 andv^t_i → λast → ∞ for alli∈Zif|1κ|<1.This is impossible because v is temporalΔ-periodic andΔ>1. Thus we know that|1κ|1.Sinceκ /0,we know thatκ−2.By4.10, we have

v_i^t

⎧⎨

⎩

v⁰_i , if tis even, i∈Z,

−v⁰_i , if tis odd, i∈Z. 4.11

Lemma 4.3. Letα, κ∈Rwithκ /0, τ, δsatisfy2.16andλ^i,ξare defined by3.2. Suppose that v{v^t_i }is aΔ,Υ-periodic traveling wave solution of 4.4with velocity−δ/τ,whereΔ>1 and Υ>1.Then the following results are true.

iFor anyt∈N,one has

v^t1₁ , . . . , v^t1_{Υ †}

1κI_Υ−αA_Υ^t1

v⁰₁ , . . . , v⁰_{Υ †}

. 4.12

iiThe vector

v₁⁰, . . . , v⁰_{Υ †}

is the sum of the vectorsuandw,whereuis an eigenvector of 1κI_Υ−αA_Υcorresponding to the eigenvalue−1 andwis either the zero vector or an eigenvector of1κI_Υ−αA_Υcorresponding to the eigenvalue 1.

iiiThe matrix1κI_Υ−αA_Υhas an eigenvalue−1,that is,1κ−α4 sin²jπ/Υ −1 for somej ∈ {0,1, . . . ,Υ/2}.

iv Δ 2.

Proof. To seei, note that the assumption on v implies

v^tΔ_i v^t_i , v^t_i v^t_iΥ ∀i, t. 4.13

Since v is a solution, by4.13, we know that

v^t1₁ , . . . , v^t1_{Υ †}

1κIΥ−αA_Υ

v^t₁ , . . . , v^t_{Υ †}

, t∈N. 4.14

By direct computation, we have

v^t1₁ , . . . , v_Υ^t1_†

1κI_Υ−αA_Υ^t1

v⁰₁ , . . . , v_Υ⁰_†

, t∈N. 4.15

Foriiandiii, by takingt1 Δin4.12, it is clear from4.13that

v⁰₁ , . . . , v_Υ⁰_†

v₁^Δ, . . . , v^Δ_{Υ †}

1κIΥ−αA_Υ^Δ

v⁰₁ , . . . , v⁰_{Υ †}

. 4.16

Thus

v⁰₁ , . . . , v_Υ⁰_†

is an eigenvector of1κIΥ−αA_Υ^Δcorresponding to the eigenvalue 1.This implies that the matrix1κI_Υ−αA_Υmust have eigenvalue 1 or−1,and

v₁⁰, . . . , v⁰_{Υ †}

uw, 4.17

whereuis either the zero vector or an eigenvector of1κIΥ−αA_Υcorresponding to the eigenvalue−1 andwis either a zero vector or an eigenvector of1κI_Υ−αA_Υcorresponding to the eigenvalue 1. Suppose that uis the zero vector, or, −1 is not an eigenvalue of1 κIΥ−αA_Υ.Then 1 must be an eigenvalue of1κIΥ−αA_Υandwmust be an eigenvector corresponding to the eigenvalue 1; otherwise,

v₁⁰, . . . , v⁰_{Υ †}

0 and this is impossible.

Thus, 1 is a temporal period of v.This is contrary toΔbeing the least among all periods and Δ>1.In conclusion,1κI_Υ−αA_Υhas eigenvalue−1 and

v₁⁰, . . . , v⁰_{Υ †}

uw,where uis an eigenvector of1κIΥ−αA_Υ corresponding to the eigenvalue−1,andw is either a zero vector or an eigenvector of1κI_Υ−αA_Υ corresponding to the eigenvalue 1.Since 1κ−αλ^0,Υ, . . . ,1κ−αλ^Υ/2,Υare all distinct eigenvalues of1κI_Υ−αA_Υ,there exists somej∈ {0,1, . . . ,Υ/2}such that1κ−αλ^j,Υ −1.

To seeiv, recall the result inii. We have

v₁⁰, . . . , v⁰_{Υ †}

1κIΥ−αA_Υ²

v⁰₁ , . . . , v_Υ⁰_†

. 4.18

It is also clear that

v^t2₁ , . . . , v^t2_{Υ †}

1κI_Υ−αA_Υ²

v^t₁ , . . . , v^t_{Υ †}

, t∈N. 4.19

That is, 2 is a temporal period of v.By the definition ofΔandΔ > 1,we haveΔ 2.The proof is complete.

Next, we consider one result about the relation betweenδandΥunder the assumption that doubly-periodic traveling wave solutions of4.4exist.

Lemma 4.4. Letα, κ ∈ Rwithκ /0 andτ, δsatisfy2.16. Suppose that v {v_i^t}is aΔ,Υ- periodic traveling wave solution of 4.4with velocity−δ/τ,whereΔ 2 andΥ≥1.

iIfτis even, thenδT₁Υfor some odd integerT₁andΥis odd.

iiIfτis odd, thenΥis even andδT₁Υ/2 for some odd integerT₁. Proof. By the assumption on v,we have

v^t2_i v^t_i , v_i^tv_iΥ^t ∀i, t. 4.20

Since v is a traveling wave, we also know that

v^tτ_i v_iδ^t ∀i, t. 4.21

To seei, suppose thatτis even. Then from4.20and4.21, we have

v^t_i v_i^t2· · ·v_i^tτv^t_iδ ∀i, t. 4.22

That is,δis also a spatial period of v.ByLemma 2.1and the definition ofΥ, it is easy to see thatδ0 mod Υ.Sinceδ0 mod Υ, τis even andτ, δ 1,we haveδT1Υfor some odd integerT₁andΥis odd.

Forii, suppose thatτ is odd.Then from4.20and4.21, we have

v^t1_i · · ·v^tτ_i v_iδ^t ∀i, t. 4.23

By4.20and4.23, we know that

v^t_i v^t2_i v_iδ^t1v^t_i2δ ∀i, t. 4.24

That is, 2δis also a spatial period of v.ByLemma 2.1and the definition ofΥ, it is easy to see that 2δ0 mod Υ.Ifδ0 mod Υ.From4.23, we have

v_i^t1v^t_iδ· · ·v_i^t ∀i, t. 4.25

Then 1 is a temporal period of v and this is contrary to Δ 2. Thusδ /0 mod Υ. Since 2δ 0 mod Υ,the fact thatΥ is odd impliesδ 0 mod Υ.This leads to a contradiction.

So we must have that Υ is even and δ 0 modΥ/2.Note that δ 0 modΥ/2and δ /0 mod ΥimpliesδT₁Υ/2 for some odd integerT₁.The proof is complete.

5. Existence Criteria

Now we turn to our main problem. First of all, letα, κ∈Rwithκ /0 andτ, δsatisfy2.16. If Υ,Δ∈ZwithΔ>1 and if4.4has aΔ,Υ-periodic traveling wave solution of4.4with velocity−δ/τ,by Lemmas4.2and4.3,Δmust be 2.For this reason, we just need to consider five mutually exclusive and exhaustive cases:i Υ Δ 1;ii Δ 1 andΥ>1;iii Δ 2 andΥ 1;iv Δ 2 andΥ 2 andv Δ 2 andΥ≥3.

The conditionΥ Δ 1 is easy to handle.

Theorem 5.1. Let α, κ ∈ R with κ /0 and τ, δ satisfy 2.16. Then the unique 1,1-periodic traveling wave solution of 4.4is{v^t_i 0}.

Proof. If v{v^t_i }is a1,1-periodic traveling wave solution of4.4, thenv_i^tcfor alli∈Z andt∈N,wherec∈R.Substituting{v_i^t c}into4.4, we havec0.Conversely, it is clear that{v_i^t 0}is a1,1-periodic traveling wave solution.

Theorem 5.2. Letα, κ∈Rwithκ /0 andτ, δsatisfy2.16. Letλ^i,ξandu^i,ξbe defined by3.2 and3.4, respectively. Then the following results hold.

iFor any Δ 1 and any Υ ≥ 2,4.4has a 1,Υ-periodic traveling wave solutions of 2.2with velocity −δ/τ if, and only if,δ 0 mod Υ, and κ−αλ^j,Υ 0 for some j ∈ {1, . . . ,Υ/2}withj,Υ 1.

iiEvery1,Υ-periodic traveling wave solution v{v^t_i }is of the form

v^t_i

v_i⁰

i∈Zu ∀t∈N, 5.1

whereu au^jbu^Υ−j for somea, b ∈ Rsuch thatau^jbu^Υ−j is a nonzero vector, and the converse is true.

Proof. Fori, let v{v_i^t}be a1,Υ-periodic traveling wave solution of2.2with velocity

−δ/τ.From the assumption on v,we have {v^t1_i }_i∈Z {v^t_i }_i∈Zfor allt ∈ N andΥis the least spatial period. Hence given anyt ∈ N,it is easy to see that the extension {v_i^t}_i∈Z of v^t₁ , . . . , v_Υ^tisΥ-periodic. Note that we also have

v^t1_i v^t_i , v_i^tv_iΥ^t ∀i, t. 5.2

Since v is a traveling wave, from5.2, we know that

v^t_iδv_i^tτ v_i^tτ−1· · ·v^t_i ∀i, t. 5.3

Therefore, given anyt∈N, δis a period of{v_i^t}_i∈Z. ByLemma 2.1, we haveδ0 mod Υ.

By Lemma 4.1, we also know that κI_Υ −αA_Υ is not invertible and v⁰₁ , . . . , v⁰_Υ is a nonzero vector in kerκI_Υ − αA_Υ. Note that κ − αλ^0,Υ, . . . , κ − αλ^Υ/2,Υ are all

distinct eigenvalues of κI_Υ − αA_Υ with corresponding eigenspaces span{u^Υ,Υ}, . . . , span{u^Υ/2,Υ, u^Υ−Υ/2,Υ},respectively. Sinceκ−αλ^0,Υκ /0 andκI_Υ−αA_Υis not invertible, we haveκ−αλ^j,Υ0 for somej ∈ {1, . . . ,Υ/2}.Hence kerκIΥ−αAΥ span{u^j,Υ, u^Υ−j,Υ} and it is clear that

v⁰₁ , . . . , v_Υ⁰_†

au^j,Υbu^Υ−j,Υ, 5.4

wherea, b ∈ Rsuch that au^j,Υbu^Υ−j,Υis a nonzero vector. If Υ 2,we see thatj must be 1 sincej ∈ {1, . . . ,Υ/2}.It is clear thatj,Υ 1,2 1.SupposeΥ≥3 and recall that the extension{v_i⁰}_i∈Z ofv⁰₁ , . . . , v⁰_Υ isΥ-periodic. ByLemma 3.1ii,the extension{v⁰_i } ofv₁⁰, . . . , v⁰_Υ isΥ-periodic if and only ifj,Υ 1.

Conversely, supposeδ 0 modΥ; there exists somej ∈ {1, . . . ,Υ/2}such thatκ− αλ^j,Υ 0 andj,Υ 1 whenΥ≥ 3. Let v {v_i^t}satisfy5.1. By the definition of v,it is clear that v is temporal 1-periodic andΥis a spatial period of v. SupposeΥ 2 and then we have thatu abu¹.The fact thatuis not a zero vector impliesab /0.ByLemma 3.1iii, we have thatuis 2-periodic. By5.1, it is clear that v is spatial 2-periodic. SupposeΥ ≥ 3.

Sincej,Υ 1,byLemma 3.1ii, we haveuisΥ-periodic. By5.1again, it is also clear that v is spatialΥ-periodic. In conclusion, we have that v is spatialΥ-periodic, that is,

v_iΥ^t v_i^t ∀t∈N, i∈Z. 5.5

Sinceu ∈ kerκIΥ−αA_Υ,from the definition of v,it is easy to check that v is a solution of 4.4. Finally, sinceδ0 mod Υ,by5.1and5.5, we know that

v^tτ_i · · ·v_i^t1v^t_i v^t_iΥ· · ·v_iδ^t; 5.6 that is, v is traveling wave with velocity−δ/τ.

To seeii, note that from the second part of the proof ini, it is easy to see that any v {v^t_i }satisfying5.1is a1,Υ-periodic traveling wave solution of4.4with velocity−δ/τ.

Also, by the first part of the proof ini, the converse is also true. The proof is complete.

We remark that any 1,Υ-periodic traveling wave solution v {v^t_i } of 4.4 is a standing wave since this v is also a traveling wave with velocity 0, that is,v^t1_i v_i^tfor all i∈Zandt∈N.

Theorem 5.3. LetΥ 1,Δ 2, α, κ∈Rwithκ /0 andτ, δsatisfy2.16.Then

i 4.4has aΔ,Υ-periodic traveling wave solution with velocity−δ/τif, and only if,κ

−2 andτis even;

iifurthermore, every such solution v{v_i^t}is of the form

v_i^t

⎧⎨

⎩

c, iftis even, i∈Z,

−c, iftis odd, i∈Z. 5.7

wherec /0,and the converse is true.