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

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

Since is a traveling wave, from (5.2), we know that

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

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,

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

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

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

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

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,

Since is even by (5.10), it is clear that

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

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

where with By Lemma 4.3(i)**,** we have

From (5.13) and (5.14), it is clear that

Since is spatial -periodic, we see that

where and From our assumption on we have

Since and are both odd, by (5.17), we have

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

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

where

for some such that is a nonzero vector, and the converse is true; while if , every such solution is of the form

where

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

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

where and is a nonzero vector. By Lemma 4.3(i), we also see that

Hence it is clear that

Now we want to show that and satisfies condition (a) or (b). First, we may assume that By (5.27), we have

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

where

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:

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

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

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

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

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.