- Research Article
- Open access
- Published:
Doubly Periodic Traveling Waves in a Cellular Neural Network with Linear Reaction
Advances in Difference Equations volume 2009, Article number: 243245 (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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F243245/MediaObjects/13662_2009_Article_1175_Equ1_HTML.gif)
or
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F243245/MediaObjects/13662_2009_Article_1175_Equ2_HTML.gif)
If we set for any
and
then it is easy to check that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F243245/MediaObjects/13662_2009_Article_1175_Equ3_HTML.gif)
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)?
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F243245/MediaObjects/13662_2009_Article_1175_Equ4_HTML.gif)
If we now assume further that a control or reaction mechanism is imposed, a slightly more complicated nonhomogeneous model such as the following
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F243245/MediaObjects/13662_2009_Article_1175_Equ5_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F243245/MediaObjects/13662_2009_Article_1175_Equ6_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F243245/MediaObjects/13662_2009_Article_1175_Equ7_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F243245/MediaObjects/13662_2009_Article_1175_Equ8_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F243245/MediaObjects/13662_2009_Article_1175_Equ9_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F243245/MediaObjects/13662_2009_Article_1175_Equ10_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F243245/MediaObjects/13662_2009_Article_1175_Equ11_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F243245/MediaObjects/13662_2009_Article_1175_Equ12_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F243245/MediaObjects/13662_2009_Article_1175_Equ13_HTML.gif)
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,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F243245/MediaObjects/13662_2009_Article_1175_Equ14_HTML.gif)
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:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F243245/MediaObjects/13662_2009_Article_1175_Equ15_HTML.gif)
if, and only if, is a
-periodic traveling wave solution with velocity
for the following equation
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F243245/MediaObjects/13662_2009_Article_1175_Equ16_HTML.gif)
Therefore, from now on, we assume in (2.2) that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F243245/MediaObjects/13662_2009_Article_1175_Equ17_HTML.gif)
where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F243245/MediaObjects/13662_2009_Article_1175_Equ18_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F243245/MediaObjects/13662_2009_Article_1175_Equ19_HTML.gif)
3. Basic Facts
Some additional basic facts are needed. Let us state these as follows. First, let be a circulant matrix defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F243245/MediaObjects/13662_2009_Article_1175_Equ20_HTML.gif)
Second, we set
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F243245/MediaObjects/13662_2009_Article_1175_Equ21_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F243245/MediaObjects/13662_2009_Article_1175_Equ22_HTML.gif)
It is known (see, e.g., [3]) that for any the eigenvalues of
are
and the eigenvector corresponding to
is
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F243245/MediaObjects/13662_2009_Article_1175_Equ23_HTML.gif)
and that are orthonormal. It is also clear that
, and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F243245/MediaObjects/13662_2009_Article_1175_Equ24_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F243245/MediaObjects/13662_2009_Article_1175_Equ25_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F243245/MediaObjects/13662_2009_Article_1175_Equ26_HTML.gif)
where such that
and
are both nonzero vectors. Let
be the extension of
so that
for
Then it is clear that for any
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F243245/MediaObjects/13662_2009_Article_1175_Equ27_HTML.gif)
By direct computation, we also have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F243245/MediaObjects/13662_2009_Article_1175_Equ28_HTML.gif)
By (3.8) and (3.9), we see that is a period of
that is,
for all
if, and only if, given any
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F243245/MediaObjects/13662_2009_Article_1175_Equ29_HTML.gif)
By (3.3) and (3.5), we may rewrite (3.10) as
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F243245/MediaObjects/13662_2009_Article_1175_Equ30_HTML.gif)
By (3.3) again, we have for each
Hence we see that
is a period of
if, and only if,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F243245/MediaObjects/13662_2009_Article_1175_Equ31_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F243245/MediaObjects/13662_2009_Article_1175_Equ32_HTML.gif)
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,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F243245/MediaObjects/13662_2009_Article_1175_Equ33_HTML.gif)
Suppose If
for some
then we have
because
Hence we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F243245/MediaObjects/13662_2009_Article_1175_Equ34_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F243245/MediaObjects/13662_2009_Article_1175_Equ35_HTML.gif)
where such that
is a nonzero vector. Define
by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F243245/MediaObjects/13662_2009_Article_1175_Equ36_HTML.gif)
(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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F243245/MediaObjects/13662_2009_Article_1175_Equ37_HTML.gif)
where such that
is a nonzero vector. Let
be defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F243245/MediaObjects/13662_2009_Article_1175_Equ38_HTML.gif)
(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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F243245/MediaObjects/13662_2009_Article_1175_Equ39_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F243245/MediaObjects/13662_2009_Article_1175_Equ40_HTML.gif)
For any it is clear that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F243245/MediaObjects/13662_2009_Article_1175_Equ41_HTML.gif)
Since is odd and
is even, by (3.22), it is easy to see that
for all
By the definition of
we also have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F243245/MediaObjects/13662_2009_Article_1175_Equ42_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F243245/MediaObjects/13662_2009_Article_1175_Equ43_HTML.gif)
If for all
and
it is clear that
for all
. By (3.21) and (3.24), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F243245/MediaObjects/13662_2009_Article_1175_Equ44_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F243245/MediaObjects/13662_2009_Article_1175_Equ45_HTML.gif)
Suppose that is a
-periodic traveling wave solutions of (4.1) with velocity
. For any
it is clear that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F243245/MediaObjects/13662_2009_Article_1175_Equ46_HTML.gif)
Then we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F243245/MediaObjects/13662_2009_Article_1175_Equ47_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F243245/MediaObjects/13662_2009_Article_1175_Equ48_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F243245/MediaObjects/13662_2009_Article_1175_Equ49_HTML.gif)
Since satisfies (4.4), by (4.5), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F243245/MediaObjects/13662_2009_Article_1175_Equ50_HTML.gif)
This fact implies that is a vector in
If
is invertible or
, by direct computation, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F243245/MediaObjects/13662_2009_Article_1175_Equ51_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F243245/MediaObjects/13662_2009_Article_1175_Equ52_HTML.gif)
Proof.
From the assumption of we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F243245/MediaObjects/13662_2009_Article_1175_Equ53_HTML.gif)
Note that also satisfies (4.4). Hence by (4.9) and computation, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F243245/MediaObjects/13662_2009_Article_1175_Equ54_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F243245/MediaObjects/13662_2009_Article_1175_Equ55_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F243245/MediaObjects/13662_2009_Article_1175_Equ56_HTML.gif)
(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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F243245/MediaObjects/13662_2009_Article_1175_Equ57_HTML.gif)
Since is a solution, by (4.13), we know that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F243245/MediaObjects/13662_2009_Article_1175_Equ58_HTML.gif)
By direct computation, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F243245/MediaObjects/13662_2009_Article_1175_Equ59_HTML.gif)
For (ii) and (iii), by taking in (4.12), it is clear from (4.13) that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F243245/MediaObjects/13662_2009_Article_1175_Equ60_HTML.gif)
Thus is an eigenvector of
corresponding to the eigenvalue
This implies that the matrix
must have eigenvalue
or
and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F243245/MediaObjects/13662_2009_Article_1175_Equ61_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F243245/MediaObjects/13662_2009_Article_1175_Equ62_HTML.gif)
It is also clear that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F243245/MediaObjects/13662_2009_Article_1175_Equ63_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F243245/MediaObjects/13662_2009_Article_1175_Equ64_HTML.gif)
Since is a traveling wave, we also know that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F243245/MediaObjects/13662_2009_Article_1175_Equ65_HTML.gif)
To see (i), suppose that is even. Then from (4.20) and (4.21), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F243245/MediaObjects/13662_2009_Article_1175_Equ66_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F243245/MediaObjects/13662_2009_Article_1175_Equ67_HTML.gif)
By (4.20) and (4.23), we know that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F243245/MediaObjects/13662_2009_Article_1175_Equ68_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F243245/MediaObjects/13662_2009_Article_1175_Equ69_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F243245/MediaObjects/13662_2009_Article_1175_Equ70_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F243245/MediaObjects/13662_2009_Article_1175_Equ71_HTML.gif)
Since is a traveling wave, from (5.2), we know that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F243245/MediaObjects/13662_2009_Article_1175_Equ72_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F243245/MediaObjects/13662_2009_Article_1175_Equ73_HTML.gif)
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,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F243245/MediaObjects/13662_2009_Article_1175_Equ74_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F243245/MediaObjects/13662_2009_Article_1175_Equ75_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F243245/MediaObjects/13662_2009_Article_1175_Equ76_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F243245/MediaObjects/13662_2009_Article_1175_Equ77_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F243245/MediaObjects/13662_2009_Article_1175_Equ78_HTML.gif)
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,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F243245/MediaObjects/13662_2009_Article_1175_Equ79_HTML.gif)
Since is even
by (5.10), it is clear that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F243245/MediaObjects/13662_2009_Article_1175_Equ80_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F243245/MediaObjects/13662_2009_Article_1175_Equ81_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F243245/MediaObjects/13662_2009_Article_1175_Equ82_HTML.gif)
where with
By Lemma 4.3(i), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F243245/MediaObjects/13662_2009_Article_1175_Equ83_HTML.gif)
From (5.13) and (5.14), it is clear that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F243245/MediaObjects/13662_2009_Article_1175_Equ84_HTML.gif)
Since is spatial
-periodic, we see that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F243245/MediaObjects/13662_2009_Article_1175_Equ85_HTML.gif)
where and
From our assumption on
we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F243245/MediaObjects/13662_2009_Article_1175_Equ86_HTML.gif)
Since and
are both odd, by (5.17), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F243245/MediaObjects/13662_2009_Article_1175_Equ87_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F243245/MediaObjects/13662_2009_Article_1175_Equ88_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F243245/MediaObjects/13662_2009_Article_1175_Equ89_HTML.gif)
where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F243245/MediaObjects/13662_2009_Article_1175_Equ90_HTML.gif)
for some such that
is a nonzero vector, and the converse is true; while if
, every such solution
is of the form
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F243245/MediaObjects/13662_2009_Article_1175_Equ91_HTML.gif)
where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F243245/MediaObjects/13662_2009_Article_1175_Equ92_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F243245/MediaObjects/13662_2009_Article_1175_Equ93_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F243245/MediaObjects/13662_2009_Article_1175_Equ94_HTML.gif)
where and
is a nonzero vector. By Lemma 4.3(i), we also see that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F243245/MediaObjects/13662_2009_Article_1175_Equ95_HTML.gif)
Hence it is clear that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F243245/MediaObjects/13662_2009_Article_1175_Equ96_HTML.gif)
Now we want to show that and
satisfies condition (a) or (b). First, we may assume that
By (5.27), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F243245/MediaObjects/13662_2009_Article_1175_Equ97_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F243245/MediaObjects/13662_2009_Article_1175_Equ98_HTML.gif)
where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F243245/MediaObjects/13662_2009_Article_1175_Equ99_HTML.gif)
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:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F243245/MediaObjects/13662_2009_Article_1175_Equ100_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F243245/MediaObjects/13662_2009_Article_1175_Equ101_HTML.gif)
Since for some odd integer
and
is odd, by (5.32), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F243245/MediaObjects/13662_2009_Article_1175_Equ102_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F243245/MediaObjects/13662_2009_Article_1175_Equ103_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F243245/MediaObjects/13662_2009_Article_1175_Equ104_HTML.gif)
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:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F243245/MediaObjects/13662_2009_Article_1175_Equ105_HTML.gif)
For any and
we define
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F243245/MediaObjects/13662_2009_Article_1175_Equ106_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F243245/MediaObjects/13662_2009_Article_1175_Equ107_HTML.gif)
We want to find all -periodic traveling wave solutions of (6.3) with velocity
By direct computation,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F243245/MediaObjects/13662_2009_Article_1175_Equ108_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F243245/MediaObjects/13662_2009_Article_1175_Equ109_HTML.gif)
where as well as
for some
with
and the converse is true. Recall that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F243245/MediaObjects/13662_2009_Article_1175_Equ110_HTML.gif)
where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F243245/MediaObjects/13662_2009_Article_1175_Equ111_HTML.gif)
In Figure 2, we take for illustration.
Example 6.3.
Let and
Set
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F243245/MediaObjects/13662_2009_Article_1175_Equ112_HTML.gif)
Consider the equation
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F243245/MediaObjects/13662_2009_Article_1175_Equ113_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F243245/MediaObjects/13662_2009_Article_1175_Equ114_HTML.gif)
where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F243245/MediaObjects/13662_2009_Article_1175_Equ115_HTML.gif)
for some such that
and
are both nonzero, and the converse is true. Recall that
where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F243245/MediaObjects/13662_2009_Article_1175_Equ116_HTML.gif)
for and
In Figure 3, we take
for illustration.
Example 6.4.
Let and
Consider the equation
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F243245/MediaObjects/13662_2009_Article_1175_Equ117_HTML.gif)
where We want to find all
-periodic traveling wave solutions of (6.9) with velocity
By direct computation, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F243245/MediaObjects/13662_2009_Article_1175_Equ118_HTML.gif)
where By direct computation again, we also know that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F243245/MediaObjects/13662_2009_Article_1175_Equ119_HTML.gif)
where
First, let with
By Theorem 5.7(i), the fact that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F243245/MediaObjects/13662_2009_Article_1175_Equ120_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F243245/MediaObjects/13662_2009_Article_1175_Equ121_HTML.gif)
where and
for some
with
and the converse is true. Recall that
where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F243245/MediaObjects/13662_2009_Article_1175_Equ122_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F243245/MediaObjects/13662_2009_Article_1175_Equ123_HTML.gif)
or
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F243245/MediaObjects/13662_2009_Article_1175_Equ124_HTML.gif)
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.
References
Szekely G: Logical networks for controlling limb movements in Urodela. Acta Physiologica Hungarica 1965, 27: 285-289.
Cheng SS, Lin JJ: Periodic traveling waves in an artificial neural network. Journal of Difference Equations and Applications 2009,15(10):963-999. 10.1080/10236190802350631
Cheng SS, Chen C-W, Wu TY: Linear time discrete periodic diffusion networks. In Differences and Differential Equations, Fields Institute Communications. Volume 42. American Mathematical Society, Providence, RI, USA; 2004:131-151.
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Lin, J., Cheng, S. Doubly Periodic Traveling Waves in a Cellular Neural Network with Linear Reaction. Adv Differ Equ 2009, 243245 (2009). https://doi.org/10.1155/2009/243245
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DOI: https://doi.org/10.1155/2009/243245