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.
-periodic vector sequence and subsequently in terms of a "doubly periodic traveling wave solution" of the dynamic bipolar cellular neural network.
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
for any 
and 
then it is easy to check that
may be called a "doubly periodic traveling wave" (see Figure 
? 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)?
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
is a function and 
is known, then we may calculate successively the sequence
of (2.2). Motivated by our example above, we want to find solutions that satisfy
and 
It is clear that equations in (1.3) are special cases of (2.4), (2.5), and (2.6), respectively.
is a double sequence satisfying (2.4) for some 
and 
Then it is clear that
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
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
and 
our traveling wave is also called a standing wave.
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.
is 
-periodic and 
is a period of 
then 
is a factor of 
or 
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.
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 (2.2) can be selected in many different ways. But naturally, we should start with the trivial polynomial and general polynomials of the form
are real numbers, and 
is a real parameter. In [
are considered. In this paper, we will consider the linear case, namely,
are mutually distinct and 
will be considered in a subsequent paper (for the important reason that quite distinct techniques are needed).
A further simplification of (2.11) is possible in view of the following translation invariance.
with 
and 
with 
Then 
is a 
-periodic traveling wave solution with velocity 
for the following equation: 
is a 
-periodic traveling wave solution with velocity 
for the following equation 
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 
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 
be a circulant matrix defined by
the eigenvalues of 
are 
and the eigenvector corresponding to 
is
are orthonormal. It is also clear that 
, and
are all distinct eigenvalues of 
with corresponding eigenspaces 
respectively.
(or vector 
), where 
, its (periodic) extension is the sequence 
defined by
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.
with 
and let 
be defined by (3.4).
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 
Let 
and 
such that 
is a nonzero vector. Then 
is 
-periodic if and only if 
Let 
such that 
Then 
is 
-periodic.
(b) 
is odd, 
and 
(c) 
is odd, 
and 
(d) 
is even, 
and 
(e) 
is even, 
and 
such that 
and 
are both nonzero vectors. Let 
be the extension of 
so that 
for 
Then it is clear that for any 
is a period of 
that is, 
for all 
if, and only if, given any 
for each 
Hence we see that 
is a period of 
if, and only if, 
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 
is a period of 
if, and only if, 
and 
.
As in (i), we also know that 
is a period of 
where 
if and only if 
That is,
If 
for some 
then we have 
because 
Hence we have 
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 
and 
and checking that 
is truly 
-periodic. The proof is complete.
be defined by (3.4). Let 
and 
with 
. Let further 
such that 
is a nonzero vector. Define 
by 
and 
is a nonzero vector. Then 
is spatial 
-periodic if, and only if, 
or 
for any 
with 
and 
is a nonzero vector. Then 
is spatial 
-periodic if, and only if, 
is a zero vector. Then 
is spatial 
-periodic if, and only if, 
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 
be even with 
and let 
be defined by (3.4). Let 
and 
such that 
is a nonzero vector. Let 
be defined by 
is a nonzero vector. Then 
for all 
and 
if and only if 
is odd and 
is even.
is a zero vector. Then 
for all 
and 
if and only if 
is odd.
is odd and 
is even. Note that 
it is clear that 
is odd and 
is even, by (3.22), it is easy to see that 
for all 
By the definition of 
we also have 
for all 
and 
is even or 
is odd. We first focus on the case that 
and 
are both even. By (3.20) and (3.22), we have
for all 
and 
it is clear that 
for all 
. By (3.21) and (3.24), we have 
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.
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).
for all 
Then we may rewrite (2.2) as
is a 
-periodic traveling wave solutions of (4.1) with velocity 
. For any 
it is clear that
-periodic traveling wave solutions of (4.1) with velocity 
do not exist.
and focus on the equation
-periodic traveling wave solutions of (4.4) with velocity 
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 
be a 
-periodic traveling wave solution of (2.2) with velocity 
It is clear that 
satisfies (4.4), by (4.5), we have 
is a vector in 
If 
is invertible or 
, by direct computation, we have 
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.
with 
and 
satisfy (2.16). Suppose that 
is a 
-periodic traveling wave solution of (4.4) with velocity 
where 
and 
Then 
each 
and 
we have 
also satisfies (4.4). Hence by (4.9) and computation, we have 
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 
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.
one has
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 
has an eigenvalue 
that is, 
for some 
.
implies 
is a solution, by (4.13), we know that 
in (4.12), it is clear from (4.13) that
is an eigenvector of 
corresponding to the eigenvalue 
This implies that the matrix 
must have eigenvalue 
or 
and 
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 
is a temporal period of 
By the definition of 
and 
we have 
The proof is complete.
and 
under the assumption that doubly-periodic traveling wave solutions of (4.4) exist.
with 
and 
satisfy (2.16). Suppose that 
is a 
-periodic traveling wave solution of (4.4) with velocity 
where 
and 
is even, then 
for some odd integer 
and 
is odd.
is odd, then 
is even and 
for some odd integer 
we have 
is a traveling wave, we also know that 
is even. Then from (4.20) and (4.21), we have
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.
is odd
Then from (4.20) and (4.21), we have
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 
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.
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 
is easy to handle.
with 
and 
satisfy (2.16). Then the unique 
-periodic traveling wave solution of (4.4) is 
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.
with 
and 
satisfy (2.16). Let 
and 
be defined by (3.2) and (3.4), respectively. Then the following results hold.
and any 
(4.4) has a 
-periodic traveling wave solutions of (2.2) with velocity 
if, and only if, 
and 
for some 
with 
-periodic traveling wave solution 
is of the form
for some 
such that 
is a nonzero vector, and the converse is true.
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 
is a traveling wave, from (5.2), we know that 
is a period of 
. By Lemma 2.1, we have 
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
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 
.
; 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,
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 
is traveling wave with velocity 
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.
-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 
with 
and 
satisfy (2.16)
Then
-periodic traveling wave solution with velocity 
if, and only if, 
and 
is even;
is of the form
and the converse is true.
be a 
-periodic traveling wave solution of (4.4) with velocity 
By Lemma 4.2, we have each 
, and 
is even. Suppose to the contrary that 
is odd. Since 
is a spatial period of 
and 
is a traveling wave, we have 
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, 
is even
by (5.10), it is clear that 
of the form (5.7) is a solution we want and the converse is also true by Lemma 4.2. The proof is complete.
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.
with 
and 
satisfy (2.16) with even 
. Then (4.4) has no 
-periodic traveling wave solutions with velocity 
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 
with 
and 
satisfy (2.16) with odd 
. Then the following results hold.
is even, then (4.4) has no 
-periodic traveling wave solutions with velocity 
is odd, 
and 
then (4.4) has no 
-periodic traveling wave solutions with velocity 
is odd, 
and 
then (4.4) has no 
-periodic traveling wave solutions with velocity 
is odd, 
and 
then any 
of the form 
and 
with 
is a 
-periodic traveling wave solution with velocity 
and the converse is true.
is odd, 
and 
then (4.4) has no 
-periodic traveling wave solutions with velocity 
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.
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
with 
By Lemma 4.3(i), we have 
is spatial 
-periodic, we see that 
and 
From our assumption on 
we have 
and 
are both odd, by (5.17), we have 
is of form (5.16) and satisfies (5.18), we have 
that is, 
This is contrary to 
The proof is complete.
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
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 
and 
Then we know that 
is not an eigenvalue of 
By Lemma 4.3(iii), 
-periodic traveling wave solutions with velocity 
do not exist.
and 
Let 
satisfy (2.16), and 
with 
Depending on the parity of the number 
, we have the following two subcases:
with 
and 
satisfy (2.16) with odd 
with 
and 
satisfy (2.16) with even 
Furthermore, when 
is odd, the conclusions in Lemma 3.3 will be used to check whether a double sequence 
is a traveling wave.
since 
Depending on whether 
for some even 
we have the following two theorems.
, and 
satisfy (C-1) above and let 
and 
be defined by (3.2) and (3.4), respectively. Suppose 
for some even 
Then
-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 
;
, every such solution 
is of the form
such that 
is a nonzero vector, and the converse is true; while if 
, every such solution 
is of the form 
such that 
is a nonzero vector, and the converse is true.
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 
and (5.24), we know that 
and 
are eigenspaces of 
corresponding to the eigenvalues 
and 
respectively. By Lemma 4.3 (ii), we have 
and 
is a nonzero vector. By Lemma 4.3(i), we also see that 
and 
satisfies condition (a) or (b). First, we may assume that 
By (5.27), we have 
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 
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: 
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).
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
for some odd integer 
and 
is odd, by (5.32), we have 
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.
is a 
-periodic traveling wave solution with velocity 
. From the proof in (i), we have shown that
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).
, and 
satisfy 
above and let 
and 
be defined by (3.2) and (3.4), respectively. Suppose 
for all even 
Then
-periodic traveling wave solution with velocity 
if, and only if, 
is even, 
for some odd integer 
and there exists some 
with 
such that 
;
is of the form 
and 
for some 
such that 
and the converse is true.
Depending on whether 
for some 
we also have the following theorems.
, and 
satisfy (C-2) above and let 
and 
be defined by (3.2) and (3.4), respectively. Suppose 
for some 
Then
- 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 
;
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.
, and 
satisfy (C-2) above and 
and let 
be defined by (3.2) and (3.4) respectively. Suppose 
for all 
Then
-periodic traveling wave solution with velocity 
if, and only if, 
is odd, 
for some odd integer 
and there exists some 
with 
such that 
; and
is of the form (5.35), and the converse is true.
and 
where 
with 
and the reaction-diffusion network:
and 
we define
is defined by (3.2). By theorems in Section 5, it is then easy to see the following result.
and 
with 
.
is the unique 
-periodic traveling wave solution of (6.1) with velocity 
for arbitrary 
and 
satisfying (2.16).
where 
with 
Then (6.1) has at least one 
-periodic traveling wave solution with velocity 
for arbitrary 
and 
which satisfy (2.16) and 
Then (6.1) has at least one 
-periodic traveling wave solution with velocity 
for arbitrary 
and 
satisfying (2.16)
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).
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 
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 
.
, and 
Consider the equation 
-periodic traveling wave solutions of (6.3) with velocity 
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 
as well as 
for some 
with 
and the converse is true. Recall that 
for illustration. 
-periodic traveling wave solution with velocity
and 
Set 
-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 
such that 
and 
are both nonzero, and the converse is true. Recall that 
where 
and 
In Figure 
for illustration. 
-periodic traveling wave solution with velocity
and 
Consider the equation 
We want to find all 
-periodic traveling wave solutions of (6.9) with velocity 
By direct computation again, we also know that 
with 
By Theorem 5.7(i), the fact that
-periodic traveling wave solutions of (6.13) with velocity 
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
and 
for some 
with 
and the converse is true. Recall that 
where 
and 
In Figure 
and 
for illustration.
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.
-periodic traveling wave solutions with velocity 
for either
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.
-periodic traveling wave solution with velocity
