We use the bifurcation method of dynamical systems to
study the periodic wave solutions and their limit forms for the KdV-like equation

Many authors have investigated the KdV-like equation

Zhang and Ma [

Recently, many authors have presented some useful methods to deal with the problems in equations, for instance [

In this paper, we use the bifurcation method mentioned above to study the periodic wave solutions for (

In Section

In this section, we state our main results for (

When

The locations of the lines

When

Using the lines and regions in Figure

For arbitrary given constant

When

When

When

When

When

When

When

When

When

When

When

When

The limiting precess of

for

for

for

The limiting precess of

for

for

for

The limiting precess of

for

for

for

The limiting precess of

for

for

for

The limiting precess of

for

for

for

Note that if

(1) When

(2) When

(3) When

In the given parametric regions, the solutions

In this section, we state our main results for (

When

The locations of the rays

When

Using the rays and regions above, we state our results as follows.

For given parameter

When

When

When

When

When

When

When

When

When

When

When

When

Similar to the reason in Remark

When

When

When

In the given regions, the solutions

In order to derive the Proposition

Integrating (

Letting

Obviously, system (

Let

Let

Using the qualitative analysis of dynamical systems, we obtain the bifurcation phase portraits of system (

When

When

It is easy to test that the closed orbit passing

When

When

When

In this section, we give derivation on Proposition

Integrating (

Letting

It is easy to see that system (

Let

Similarly, using the qualitative analysis of dynamical systems, we get the bifurcation phase portraits of system (

When

When

It is easy to test that the closed orbit passing

When

When

When

In this paper, Using the special closed orbits, we have obtained trigonometric function periodic wave solutions for (

Now, we point out that the trigonometric function periodic wave solutions can be obtained from the limits of the elliplic function periodic wave solution. For given real number

Assume that

There are two closed orbits

The locations of

On

Substituting (

Solving (

Letting

Therefore, in (

Via Remark

We also have tested the correctness of each solution by using the software Mathematica. Here, we list two testing orders. Others testing orders are similar.

The orders for testing

The orders for testing

Research is supported by the National Natural Science Foundation of China (no. 10871073) and the Research Expences of Central Universities for students.