The F-expansion method is used to find traveling wave solutions to various wave equations. By giving more solutions of the general subequation, an extended F-expansion method is introduced by Emmanuel. In our work, a generalized KdV type equation of neglecting the highest-order infinitesimal term, which is an important water wave model, is discussed by using the extended F-expansion method. And when the parameters satisfy certain relations, some new exact solutions expressed by Jacobi elliptic function, hyperbolic function, and trigonometric function are obtained. The related results are enriched.
It has recently become more interesting to obtain exact solutions of nonlinear partial differential equations. These equations are mathematical models of complex physical phenomena that arise in engineering, applied mathematics, chemistry, biology, mechanics, physics, and so forth. Thus, the investigation of the traveling wave solutions to nonlinear evolution equations (NLEEs) plays an important role in mathematical physics. A lot of physical models have supported a wide variety of solitary wave solutions.
In the recent years, much effort has been spent on this task and many significant methods have been established such as inverse scattering transform [
Wang and Li [
In this work, we apply the extended F-expansion method on a higher-order wave equation of KdV type for obtaining new exact traveling solutions.
In 1995, based on the physical and asymptotic considerations, Fokas [
Assuming that the waves are unidirectional and neglecting terms of
If only we neglect the highest-order infinitesimal term of
In fact, (
Equation (
The organization of the paper is as follows. In Section
Based on F-expansion method, the main procedures of the extended F-expansion method are as follows [
Consider a general nonlinear PDE in the form
Suppose that the solution of ODE (
Substituting (
Assuming that the constants
Making a transformation
Suppose that (
Substituting (
In this situation, we have the following cases.
Consider
Consider
Also
Substituting (
Moreover,
We have
When
The solutions of (
Solutions of
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When
From (
When
From (
When
When
From (
When
When
In this situation, we have
Substituting (
When
The solutions of (
Solutions of (
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Combining (
If
In this case, there exist three parameters
Equation (
Substituting (
Consider
Consider
Consider
Substituting (
Also
Moreover,
Substituting solutions of (
When
In this case, there exist three parameters
The investigation of the exact solutions of (
The authors declare that there is no conflict of interests regarding the publication of this paper.
This research is supported by the Natural Science Foundation of China (no. 11161020 and no. 11361023) and the Natural Science Foundation of Yunnan (2013FZ117).