Open Access

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

Advances in Difference Equations20092009:243245

DOI: 10.1155/2009/243245

Received: 4 June 2009

Accepted: 13 October 2009

Published: 1 December 2009

Abstract

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.

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 different (artificial) neurons and the sequential rhythm can be explained in terms of an -periodic vector sequence and subsequently in terms of a "doubly periodic traveling wave solution" of the dynamic bipolar cellular neural network.

Similar dynamic (locomotive) patterns can be observed in many animal behaviors and therefore we need not repeat the same description in [1]. 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.

Let and 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 and downward if the state value of the corresponding neuron pool is Let and be the state values of and during the time stage where Then the movements of our legs in terms of will form a -periodic sequential pattern

(1.1)

or

(1.2)

If we set for any and then it is easy to check that

(1.3)
Such a sequence 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 ? 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)?
Figure 1

Doubly periodic traveling wave.

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 in [2] has 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 form (see the statement after (2.11)).

2. The Model

We briefly recall the diffusion-reaction network in [2]. In the following, we set and For any we also use [ ] to denote the greatest integer part of Suppose that are neuron pools, where placed (in a counterclockwise manner) on the vertices of a regular polygon such that each neuron pool has exactly two neighbors, and where For the sake of convenience, we have set and to reflect the fact that these neuron pools are placed on the vertices of a regular polygon. For the same reason, we define for any and let each be the state value of the th unit in the time period During the time period , if the value of the th unit is higher than , we assume that "information" will flow from the th unit to its neighbor. The subsequent change of the state value of the th unit is , and it is reasonable to postulate that it is proportional to the difference , say, , where is a proportionality constant. Similarly, information is assumed to flow from the -unit to the th unit if . Thus, it is reasonable that the total effect is

(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

(2.2)

may result. In the above model, we assume that is a function and

The existence and uniqueness of (real) solutions of (2.2) is easy to see. Indeed, if the (real) initial distribution is known, then we may calculate successively the sequence

(2.3)

in a unique manner, which will give rise to a unique solution of (2.2). Motivated by our example above, we want to find solutions that satisfy

(2.4)
(2.5)
(2.6)

where and It is clear that equations in (1.3) are special cases of (2.4), (2.5), and (2.6), respectively.

Suppose that is a double sequence satisfying (2.4) for some and Then it is clear that

(2.7)

where Hence when we want to find any solution of (2.2) satisfying (2.4), it is sufficient to find the solution of (2.2) satisfying

(2.8)

where is the greatest common divisor of and For this reason, we will pay attention to the condition that Formally, given any and with a real double sequence is called a traveling wave with velocity if

(2.9)

In case and our traveling wave is also called a standing wave.

Next, recall that a positive integer is called a period of a sequence if for all . Furthermore, if is the least among all periods of a sequence then is said to be -periodic. It is clear that if a sequence is periodic, then the least number of all its (positive) periods exists. It is easy to see the following relation between the least period and a period of a periodic sequence.

Lemma 2.1.

If is -periodic and is a period of then is a factor of or

We may extend the above concept of periodic sequences to double sequences. Suppose that is a real double sequence. If such that for all and then is called a spatial period of Similarly, if such that for all and then is called a temporal period of . Furthermore, if is the least among all spatial periods of , then is called spatial -periodic, and if is the least among all temporal periods of then is called temporal -periodic.

In seeking solutions of (2.2) that satisfy (2.5) and (2.6), in view of Lemma 2.1, there is no loss of generality to assume that the numbers and are 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 of (2.2). More precisely, given any function and with in this paper, we will mainly be concerned with the traveling wave solutions of (2.2) with velocity which are also spatial -periodic and temporal -periodic. For convenience, we call such solutions -periodic traveling wave solutions of (2.2) with velocity

In general, the control function in (2.2) can be selected in many different ways. But naturally, we should start with the trivial polynomial and general polynomials of the form

(2.10)

where are real numbers, and is a real parameter. In [2], the trivial polynomial and the quadratic polynomial are considered. In this paper, we will consider the linear case, namely,

(2.11)

while the cases where are mutually distinct and will be considered in a subsequent paper (for the important reason that quite distinct techniques are needed).

Since the trivial polynomial is considered in [2], we may avoid the case where A further simplification of (2.11) is possible in view of the following translation invariance.

Lemma 2.2.

Let with and with Then is a -periodic traveling wave solution with velocity for the following equation:
(2.12)
if, and only if, is a -periodic traveling wave solution with velocity for the following equation
(2.13)

Therefore, from now on, we assume in (2.2) that

(2.14)

where

(2.15)

As for the traveling wave solutions, we also have the following reflection invariance result (a direct verification is easy and can be found in [2]).

Lemma 2.3 (cf. proof of [2, Theorem  3]).

Given any and with If is a traveling wave solution of (2.2) with velocity then is also a traveling wave solution of (2.2) with velocity

Let and where Suppose that is a -periodic traveling wave solution of (2.2) with velocity Then it is easy to check that is also temporal -periodic and spatial -periodic. From this fact and Lemma 2.3, when we want to consider the -periodic traveling wave solutions of (2.2) with velocity , it is sufficient to consider the -periodic traveling wave solutions of (2.2) with velocity . In conclusion, from now on, we may restrict our attention to the case where
(2.16)

3. Basic Facts

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

(3.1)

Second, we set

(3.2)
(3.3)

It is known (see, e.g., [3]) that for any the eigenvalues of are and the eigenvector corresponding to is

(3.4)

and that are orthonormal. It is also clear that , and

(3.5)

Therefore, are all distinct eigenvalues of with corresponding eigenspaces respectively.

Given any finite sequence (or vector ), where , its (periodic) extension is the sequence defined by

(3.6)

Suppose that and satisfy (2.16). When we want to know whether a double sequence is a -periodic traveling wave solution of (2.2) with velocity the following two results will be useful.

Lemma 3.1.

Let with and let be defined by (3.4).

(i)Suppose Let with and such that and are both nonzero vectors. Then is a period of the extension of the vector if and only if and

(ii)Suppose Let and such that is a nonzero vector. Then is -periodic if and only if

(iii)Suppose Let such that Then is -periodic.

Proof.

To see (i), we need to consider five mutually exclusive and exhaustive cases: (a) (b) is odd, and (c) is odd, and (d) is even, and (e) is even, and

Suppose that case (a) holds. Take

(3.7)
where such that and are both nonzero vectors. Let be the extension of so that for Then it is clear that for any
(3.8)
By direct computation, we also have
(3.9)
By (3.8) and (3.9), we see that is a period of that is, for all if, and only if, given any
(3.10)
By (3.3) and (3.5), we may rewrite (3.10) as
(3.11)
By (3.3) again, we have for each Hence we see that is a period of if, and only if,
(3.12)
Note that implies that and are distinct and hence they are linearly independent. Thus, the fact that is not a zero vector implies Similarly, we also have Then it is easy to check that and are linear independent. Hence we have that is a period of if and only if
(3.13)

In other words, is a period of if, and only if, and .

The other cases (b)–(e) can be proved in similar manners and hence their proofs are skipped.

To prove (ii), we first set As in (i), we also know that is a period of where if and only if That is,

(3.14)
Suppose If for some then we have because Hence we have
(3.15)

In other words, if then is -periodic. Next, suppose that is, there exists some such that and Note that and hence we also have Since and we have Taking then we have Hence is a period of and That is, is not -periodic. In conclusion, if is -periodic, then

The proof of (iii) is done by recalling that and and checking that is truly -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.

Let be defined by (3.4). Let and with . Let further
(3.16)
where such that is a nonzero vector. Define by
(3.17)

(i)Suppose that and is a nonzero vector. Then is spatial -periodic if, and only if, or for any with

(ii)Suppose that and is a nonzero vector. Then is spatial -periodic if, and only if,

(iii)Suppose that is a zero vector. Then is spatial -periodic if, and only if,

Proof.

To see (i), suppose that and is a nonzero vector Note that the fact that with implies By Lemma 3.1(i), is a period of if, and only if, and . By Lemma 3.1(i)again, and if, and only if, is a period of Hence the least period of is the same as and is spatial -periodic if, and only if, is -periodic. Note that is a period of By Lemma 2.1 and Lemma 3.1(i), we have is -periodic if and only if or for any with

The assertions (ii) and (iii) can be proved in similar manners. The proof is complete.

Lemma 3.3.

Let be even with and let be defined by (3.4). Let and
(3.18)
where such that is a nonzero vector. Let be defined by
(3.19)

(i)Suppose that is a nonzero vector. Then for all and if and only if is odd and is even.

(ii)Suppose that is a zero vector. Then for all and if and only if is odd.

Proof.

To see (i), we first suppose that is odd and is even. Note that
(3.20)
(3.21)
For any it is clear that
(3.22)
Since is odd and is even, by (3.22), it is easy to see that for all By the definition of we also have
(3.23)

In particular, we have for all and

For the converse, suppose that is even or is odd. We first focus on the case that and are both even. By (3.20) and (3.22), we have

(3.24)
If for all and it is clear that for all . By (3.21) and (3.24), we have
(3.25)

That is, This is contrary to our assumption. That is, if are both even, then we have for some and By similar arguments, in case where are both odd or where is even and is odd, we also have for some and In summary, if for all then is odd and is even.

The assertion (ii) is proved in a manner similar to that of (i). The proof is complete.

4. Necessary Conditions

Let in this section, we want to find the necessary and sufficient conditions for -periodic traveling wave solutions of (2.2) with velocity under the assumptions (2.14), (2.15), and (2.16).

We first consider the case where for all Then we may rewrite (2.2) as

(4.1)

Suppose that is a -periodic traveling wave solutions of (4.1) with velocity . For any it is clear that

(4.2)

Then we have

(4.3)

This is a contradiction. In other words, -periodic traveling wave solutions of (4.1) with velocity do not exist.

Next, we consider the case and focus on the equation

(4.4)

Before dealing with this case, we give some necessary conditions for the existence of -periodic traveling wave solutions of (4.4) with velocity

Lemma 4.1.

Let with and satisfy (2.16). Suppose that is a -periodic traveling wave solution of (4.4) with velocity where and Then the matrix is not invertible and is a nonzero vector in

Proof.

Let be a -periodic traveling wave solution of (2.2) with velocity It is clear that
(4.5)
Since satisfies (4.4), by (4.5), we have
(4.6)
This fact implies that is a vector in If is invertible or , by direct computation, we have
(4.7)

and hence for all and This is contrary to being the least among all spatial periods and . That is, is not invertible and is a nonzero vector in The proof is complete.

Lemma 4.2.

Let with and satisfy (2.16). Suppose that is a -periodic traveling wave solution of (4.4) with velocity where and Then each and
(4.8)

Proof.

From the assumption of we have
(4.9)
Note that also satisfies (4.4). Hence by (4.9) and computation, we have
(4.10)
If for some by (4.9) and (4.10), we have for all and for any This is contrary to being the least among all temporal periods and Hence we have for all Then it is clear that is divergent as if and as for all if This is impossible because is temporal -periodic and . Thus we know that Since we know that By (4.10), we have
(4.11)

Lemma 4.3.

Let with satisfy (2.16) and are defined by (3.2). Suppose that is a -periodic traveling wave solution of (4.4) with velocity where and Then the following results are true.

(i)For any one has

(4.12)

(ii)The vector is the sum of the vectors and where is an eigenvector of corresponding to the eigenvalue and is either the zero vector or an eigenvector of corresponding to the eigenvalue

(iii)The matrix has an eigenvalue that is, for some .

(iv)

Proof.

To see (i), note that the assumption on implies
(4.13)
Since is a solution, by (4.13), we know that
(4.14)
By direct computation, we have
(4.15)

For (ii) and (iii), by taking in (4.12), it is clear from (4.13) that

(4.16)
Thus is an eigenvector of corresponding to the eigenvalue This implies that the matrix must have eigenvalue or and
(4.17)

where is either the zero vector or an eigenvector of corresponding to the eigenvalue and is either a zero vector or an eigenvector of corresponding to the eigenvalue 1. Suppose that is the zero vector, or, is not an eigenvalue of Then must be an eigenvalue of and must be an eigenvector corresponding to the eigenvalue 1; otherwise, and this is impossible. Thus, is a temporal period of This is contrary to being the least among all periods and In conclusion, has eigenvalue and where is an eigenvector of corresponding to the eigenvalue and is either a zero vector or an eigenvector of corresponding to the eigenvalue Since are all distinct eigenvalues of there exists some such that

To see (iv), recall the result in (ii). We have

(4.18)
It is also clear that
(4.19)

That is, is a temporal period of By the definition of and we have The proof is complete.

Next, we consider one result about the relation between and under the assumption that doubly-periodic traveling wave solutions of (4.4) exist.

Lemma 4.4.

Let with and satisfy (2.16). Suppose that is a -periodic traveling wave solution of (4.4) with velocity where and

If is even, then for some odd integer and is odd.

If is odd, then is even and for some odd integer

Proof.

By the assumption on we have
(4.20)
Since is a traveling wave, we also know that
(4.21)

To see (i), suppose that is even. Then from (4.20) and (4.21), we have

(4.22)

That is, is also a spatial period of By Lemma 2.1 and the definition of , it is easy to see that Since is even and we have for some odd integer and is odd.

For (ii), suppose that is odd Then from (4.20) and (4.21), we have

(4.23)
By (4.20) and (4.23), we know that
(4.24)
That is, is also a spatial period of By Lemma 2.1 and the definition of , it is easy to see that If From (4.23), we have
(4.25)

Then is a temporal period of and this is contrary to Thus Since the fact that is odd implies This leads to a contradiction. So we must have that is even and Note that and implies for some odd integer The proof is complete.

5. Existence Criteria

Now we turn to our main problem. First of all, let with and satisfy (2.16). If with and if (4.4) has a -periodic traveling wave solution of (4.4) with velocity by Lemmas 4.2 and 4.3, must be For this reason, we just need to consider five mutually exclusive and exhaustive cases: (i) (ii) and (iii) and (iv) and and (v) and

The condition is easy to handle.

Theorem 5.1.

Let with and satisfy (2.16). Then the unique -periodic traveling wave solution of (4.4) is

Proof.

If is a -periodic traveling wave solution of (4.4), then for all and where Substituting into (4.4), we have Conversely, it is clear that is a -periodic traveling wave solution.

Theorem 5.2.

Let with and satisfy (2.16). Let and be defined by (3.2) and (3.4), respectively. Then the following results hold.

(i)For any and any (4.4) has a -periodic traveling wave solutions of (2.2) with velocity if, and only if, and for some with

(ii)Every -periodic traveling wave solution is of the form

(5.1)

where for some such that is a nonzero vector, and the converse is true.

Proof.

For (i), let be a -periodic traveling wave solution of (2.2) with velocity From the assumption on we have for all and is the least spatial period. Hence given any it is easy to see that the extension of is -periodic. Note that we also have
(5.2)
Since is a traveling wave, from (5.2), we know that
(5.3)

Therefore, given any is a period of . By Lemma 2.1, we have

By Lemma 4.1, we also know that is not invertible and is a nonzero vector in Note that are all distinct eigenvalues of with corresponding eigenspaces respectively. Since and is not invertible, we have for some Hence and it is clear that

(5.4)

where such that is a nonzero vector. If we see that must be since It is clear that Suppose and recall that the extension of is -periodic. By Lemma 3.1(ii),the extension of is -periodic if and only if .

Conversely, suppose ; there exists some such that and when . Let satisfy (5.1). By the definition of it is clear that is temporal -periodic and is a spatial period of . Suppose and then we have that The fact that is not a zero vector implies By Lemma 3.1(iii), we have that is -periodic. By (5.1), it is clear that is spatial -periodic. Suppose Since by Lemma 3.1(ii), we have is -periodic. By (5.1) again, it is also clear that is spatial -periodic. In conclusion, we have that is spatial -periodic, that is,

(5.5)
Since from the definition of it is easy to check that is a solution of (4.4). Finally, since by (5.1) and (5.5), we know that
(5.6)

that is, is traveling wave with velocity

To see (ii), note that from the second part of the proof in (i), it is easy to see that any satisfying (5.1) is a -periodic traveling wave solution of (4.4) with velocity Also, by the first part of the proof in (i), the converse is also true. The proof is complete.

We remark that any -periodic traveling wave solution of (4.4) is a standing wave since this is also a traveling wave with velocity , that is, for all and

Theorem 5.3.

Let with and satisfy (2.16) Then

(i)(4.4) has a -periodic traveling wave solution with velocity if, and only if, and is even;

(ii)furthermore, every such solution is of the form

(5.7)

where and the converse is true.

Proof.

To see (i), let be a -periodic traveling wave solution of (4.4) with velocity By Lemma 4.2, we have each , and
(5.8)
We just need to show that is even. Suppose to the contrary that is odd. Since is a spatial period of and is a traveling wave, we have
(5.9)
This is contrary to the fact that is least among all temporal periods That is, is even. For the converse, suppose that and is even. Let be defined by (5.7). Since by the definition of , it is clear that is the least temporal period and is the least spatial period. That is,
(5.10)
Since is even by (5.10), it is clear that
(5.11)

For (ii), from the proof in (i), we know that any of the form (5.7) is a solution we want and the converse is also true by Lemma 4.2. The proof is complete.

Now we consider the case In this case, and are specific integers. Hence it is relatively easy to find the -periodic traveling wave solutions of (4.4) with velocity for any satisfying (2.16). Depending on the parity of we have two results.

Theorem 5.4.

Let with and satisfy (2.16) with even . Then (4.4) has no -periodic traveling wave solutions with velocity

Proof.

Since is even, by Lemma 4.4(i), a necessary condition for the existence of -periodic traveling wave solutions with velocity is that is odd. This is contrary to our assumption that

Theorem 5.5.

Let with and satisfy (2.16) with odd . Then the following results hold.

(i)If is even, then (4.4) has no -periodic traveling wave solutions with velocity

(ii)If is odd, and then (4.4) has no -periodic traveling wave solutions with velocity

(iii)If is odd, and then (4.4) has no -periodic traveling wave solutions with velocity

(iv)If is odd, and then any of the form
(5.12)

where and with is a -periodic traveling wave solution with velocity and the converse is true.

(v)If is odd, and then (4.4) has no -periodic traveling wave solutions with velocity

Proof.

To see (i), suppose is even. By Lemma 4.4(ii),a necessary condition for the existence of such solutions is for some odd integer Hence the fact that implies is odd. This leads to a contradiction.

For (ii), let and is odd. By direct computation, we have and are eigenvalues of with corresponding eigenvectors and respectively. Suppose is a -periodic traveling wave solution with velocity By Lemma 4.3(ii), we have

(5.13)
where with By Lemma 4.3(i), we have
(5.14)
From (5.13) and (5.14), it is clear that
(5.15)
Since is spatial -periodic, we see that
(5.16)
where and From our assumption on we have
(5.17)
Since and are both odd, by (5.17), we have
(5.18)

Since is of form (5.16) and satisfies (5.18), we have that is, This is contrary to The proof is complete.

For (iii), suppose that is odd, and Then we have that is an eigenvalue of with corresponding eigenvector and another eigenvalue Suppose that is a -periodic traveling wave solution with velocity By Lemma 4.3(ii), we have

(5.19)

Since is a spatial period of by (5.19), it is easy to see that is the least spatial period. This leads to a contradiction. Hence (4.4) has no -periodic traveling wave solutions with velocity

The assertion (iv) is proved by the same method used in (ii).

For (v), suppose and Then we know that is not an eigenvalue of By Lemma 4.3(iii), -periodic traveling wave solutions with velocity do not exist.

Finally, we consider the case where and Let satisfy (2.16), and with Depending on the parity of the number , we have the following two subcases:

(C-1) with and satisfy (2.16) with odd

(C-2) with and satisfy (2.16) with even

Here the facts in Lemma 3.2 will be used to check the spatial period of a double sequence Furthermore, when is odd, the conclusions in Lemma 3.3 will be used to check whether a double sequence is a traveling wave.

Now we focus on case (C-1). Note that since Depending on whether for some even we have the following two theorems.

Theorem 5.6.

Let , and satisfy (C-1) above and let and be defined by (3.2) and (3.4), respectively. Suppose for some even Then

(i)(4.4) has a -periodic traveling wave solution with velocity if, and only if, is even, for some odd integer and there exists some such that and either (a) or (b) is odd and for any with one has either or ;

(ii)furthermore, if , every such solution is of the form

(5.20)
where
(5.21)
for some such that is a nonzero vector, and the converse is true; while if , every such solution is of the form
(5.22)
where
(5.23)

for some such that is a nonzero vector, and the converse is true.

Proof.

Let be a -periodic traveling wave solution with velocity Since is odd, by Lemma 4.4(ii), we have that is even and for some odd integer From Lemma 4.3(iii), we also have
(5.24)
In view of and (5.24), we know that and are eigenspaces of corresponding to the eigenvalues and respectively. By Lemma 4.3 (ii), we have
(5.25)
where and is a nonzero vector. By Lemma 4.3(i), we also see that
(5.26)
Hence it is clear that
(5.27)
Now we want to show that and satisfies condition (a) or (b). First, we may assume that By (5.27), we have
(5.28)
where Under the assumption we also have that is a nonzero vector. Otherwise, is the least spatial period and this is contrary to Recall that is a spatial period of Hence by (5.28), we have
(5.29)
where
(5.30)
By Lemma 3.2(ii), is spatial -periodic if, and only if, Note that and are both even. This leads to a contradiction. In other words, we have that is, Next, we prove that satisfies condition (a) or (b). We may assume that the result is not true. In other words, we have either and is even or and for some with Under this assumption, we have . Otherwise, by (5.27), Lemma 3.2(iii),and the fact that , we know that is not spatial -periodic. This leads to a contradiction. Note that is odd, for some odd , and is a -periodic traveling wave. These facts imply that has the following property:
(5.31)

If and is even, by Lemma 3.3(i), does not satisfy (5.31). This leads to a contradiction. If and for some with by Lemma 3.2(i), we see that is not spatial -periodic. This leads to a contradiction again. In conclusion, we have that satisfies condition (a) or (b).

For the converse, suppose that is even, for some odd integer and for some We further suppose that satisfies (a) and let be defined by (5.20). Recall that and are eigenspaces of corresponding to the eigenvalues and respectively. Hence by direct computation, we have that is a solution of (4.4). Since , we also have that is temporal -periodic. Since we have for any with By (i) and (iii) of Lemma 3.2, it is easy to check that is spatial -periodic. The fact implies that is odd. From (i) and (ii) of Lemma 3.3, we have that

(5.32)
Since for some odd integer and is odd, by (5.32), we have
(5.33)

In other words, is a -periodic traveling wave solution with velocity If satisfies (b), we simply let be defined by (5.22) and then the desired result may be proved by similar arguments.

To see (ii), suppose that is a -periodic traveling wave solution with velocity . From the proof in (i), we have shown that

(5.34)

where and is a nonzero vector. Since is a spatial period of we have that is of the form (5.20). Now we just need to show that if then we have . Suppose to the contrary that is a zero vector, and By Lemma 3.2 (iii), is not spatial -periodic. This leads to a contradiction. The converse has been shown in the second part of the proof of (i).

Theorem 5.7.

Let , and satisfy above and let and be defined by (3.2) and (3.4), respectively. Suppose for all even Then

(i)(4.4) has a -periodic traveling wave solution with velocity if, and only if, is even, for some odd integer and there exists some with such that ;

(ii)furthermore, every such solution is of the form
(5.35)

where and for some such that and the converse is true.

Next, we focus on case (C-2) and recall that Depending on whether for some we also have the following theorems.

Theorem 5.8.

Let , and satisfy (C-2) above and let and be defined by (3.2) and (3.4), respectively. Suppose for some Then

(i)(4.4) has a - periodic traveling wave solution with velocity if, and only if, is odd, for some odd integer and for some such that either (a) and or (b) with or (c) with and for any with one has either or ;

(ii)furthermore, if satisfies condition (i)–(a) above, every such solution is of the form (5.20), and the converse is true; while if satisfies condition (i)–(b) above, every such solution is of the form (5.22), and the converse is true.

Theorem 5.9.

Let , and satisfy (C-2) above and and let be defined by (3.2) and (3.4) respectively. Suppose for all Then

(i)(4.4) has a -periodic traveling wave solution with velocity if, and only if, is odd, for some odd integer and there exists some with such that ; and

(ii)furthermore, every such solution is of the form (5.35), and the converse is true.

6. Concluding Remarks and Examples

Recall that one of our main concerns is whether mathematical models can be built that supports doubly periodic traveling patterns (with a priori unknown velocities and periodicities). In the previous discussions, we have found necessary and sufficient conditions for the existence of traveling waves with arbitrarily given least spatial periods and least temporal periods and traveling speeds. Therefore, we may now answer our original question as follows. Suppose that we are given the parameters and where with and the reaction-diffusion network:

(6.1)

For any and we define

(6.2)

where is defined by (3.2). By theorems in Section 5, it is then easy to see the following result.

Corollary 6.1.

Let and with .

( )The double sequence is the unique -periodic traveling wave solution of (6.1) with velocity for arbitrary and satisfying (2.16).

( )Suppose where with Then (6.1) has at least one -periodic traveling wave solution with velocity for arbitrary and which satisfy (2.16) and

( )Suppose Then (6.1) has at least one -periodic traveling wave solution with velocity for arbitrary and satisfying (2.16)

( )Suppose or and Then (6.1) has at least one -periodic traveling wave solution with velocity for arbitrary and which are both odd and satisfy (2.16).

( )Suppose (i) where is even, with or (ii) where is even, with odd and even and for any with one has either or Then (6.1) has at least one -periodic traveling wave solution with velocity for arbitrary and which satisfy (2.16), is odd, and for some odd integer

( )Suppose (i) where is odd, with or (ii) where is odd, and for any with one has either or Then (6.1) has at least one -periodic traveling wave solution with velocity for arbitrary and which satisfy (2.16), is even, and .

Finally, we provide some examples to illustrate the conclusions in the previous sections.

Example 6.2.

Let , and Consider the equation
(6.3)

We want to find all -periodic traveling wave solutions of (6.3) with velocity

By direct computation,

(6.4)
It is also clear that is even, for some odd integer , and By Theorem 5.7(i),(6.3) has -periodic traveling wave solution with velocity By Theorem 5.7(ii), any such solution of (6.3) is of the form
(6.5)
where as well as for some with and the converse is true. Recall that
(6.6)
where
(6.7)
In Figure 2, we take for illustration.
Figure 2

A -periodic traveling wave solution with velocity

Example 6.3.

Let and Set
(6.8)
Consider the equation
(6.9)
We want to find all -periodic traveling wave solutions of (6.9) with velocity By direct computation, we have From our assumption, we also have . Note that and for any with By Theorem 5.8(i), (6.9) has doubly periodic traveling wave solutions. By Theorem 5.8(ii),any solution is of the form
(6.10)
where
(6.11)
for some such that and are both nonzero, and the converse is true. Recall that where
(6.12)
for and In Figure 3, we take for illustration.
Figure 3

A -periodic traveling wave solution with velocity

Example 6.4.

Let and Consider the equation
(6.13)

where We want to find all -periodic traveling wave solutions of (6.9) with velocity

By direct computation, we have

(6.14)
where By direct computation again, we also know that
(6.15)

where

First, let with By Theorem 5.7(i), the fact that

(6.16)

is necessary for the existence of doubly periodic traveling wave solutions. From (6.14), one has that (6.13) has no -periodic traveling wave solutions of (6.13) with velocity

Secondly, let where Recall (6.15), we see that for all By our assumption, it is easy to check that is even and for some odd integer . We also have and note that because of By Theorem 5.7(i), (6.13) has doubly periodic traveling wave solutions. By Theorem 5.7(ii), any solution is of the form

(6.17)
where and for some with and the converse is true. Recall that where
(6.18)

for and In Figure 4, we take and for illustration.

Finally, let where We also have for all is even, for some odd integer, and However, it is clear that By Theorem 5.7(i),(6.13) has no doubly periodic traveling wave solutions.

We have given a complete account for the existence of -periodic traveling wave solutions with velocity for either

(6.19)
or
(6.20)
In particular, the former equation does not have any such solutions, while the latter may, but only when or We are then able to pinpoint the exact conditions on , and such that the desired solutions exist. Although we are concerned with the case where the reaction term is linear, the number of parameters involved, however, leads us to a relatively difficult problem as can be seen in our previous discussions.
Figure 4

A -periodic traveling wave solution with velocity

Authors’ Affiliations

(1)
Department of Mathematics, Tsing Hua University

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