Late in the 19th century, Korteweg and de Vries developed a theory to describe weakly nonlinear wave propagation in shallow water. The classical Korteweg-de Vries (KdV) equation is usually written as
After a long time, the KdV equation has been found to be involved in a wide range of physics phenomena, especially those exhibiting shock waves, travelling waves, and solitons. Certain theoretical physics phenomena in the quantum mechanics domain can be explained by means of KdV model.
As is well known, the classical KdV equation has been played a central role in the study of nonlinear phenomena, especially solitons phenomena which exist due to a balance between weak nonlinearity and dispersion. As one of the most fundamental equations of solitons phenomena, Equation (1) has caused great attention from many researchers, all forms of modified KdV equations have been studied extensively (see  -  ).
Tzirtzilakis, et al.  discussed second and third order approximations of water wave equations of KdV type. Analytical expression for solitary wave solutions for some special equations was derived. By using a Fourier pseudospectral method combined with a finite-difference scheme, a detailed numerical study of these solutions obtained in  was carried out. The stability of these solitary wave solutions was also established.
Rosenau and Hyman  introduced and studied a class of KdV equations— equation. They discovered that the solitary solutions of these equations, for certain m and n, have compact support, namely they vanish outside a finite core region. Solitons with finite wavelength are called compactons.
In  , Rosenau subsequently studied the model
where . This model emerged in nonlinear lattices and was used to describe the dispersion of dilute suspensions for . But Rosenau  only got general formulas in terms of the cosine for model (2). With the use of new ansatze methods, Wazwaz  examined model (2) for two cases, and . And the exact travelling solutions in terms of sine, cosine function, the hyperbolic function sinh and cosh were derived.
Wazwaz investigated variants of the KdV equations respectively in  and  as follows:
where a is a nonzero constant. The compactons and solitary pattern solutions were presented.
The present work aims to extend the work made by Wazwaz   . We desire to seek another method to solve nonlinear equations. For this purpose, the wave variable is introduced to carry the PDEs into ODEs. By using this variable replacement method, some new exact solutions including solitons can be obtained. In fact, the method in this paper is efficient to solve many nonlinear equations. It avoids tedious algebra and guesswork and also can be used in higher dimensional space.
In this paper, we will discuss generalized KdV equations, Equaiton (3) and Equaiton (4) and the following equations with negative exponents:
where a is a nonzero constant. In the sense of ignoring the constants of integration resulted from solving Equations (3)-(6), the exact travelling solutions have been obtained which contain the main results made in   as special cases.
2. The Generalized KdV Equations with Positive Exponents
2.1. Exact Travelling Wave Solutions for Equation (3)
Firstly, we assume that the travelling wave solutions of Equation (3) take the form
in which , .
Substituting (7) and (8) into Equation (3) gives the following nonlinear ODE
Integrating Equation (9) once and setting the constant of integration to be zero, we find
Considering , we get
Set , then
Letting , we get . So Equation (12) becomes
By using the separating variants method, we have
Case 1. : Solving Equation (15) gives
Hence, we limit the domain of , obtain the following compacton solutions:
Case 2. : Solving Equation (15), we get the solitary pattern solutions as follows:
Remark 1. Letting in (18) and (19), we have
which just are the main results for Equation (3) obtained by Wazwaz  . In other words, solutions (22), (23) made in  are special cases of formulas (18), (19).
2.2. Exact Travelling Wave Solutions for Equation (4)
Following the analysis presented above, we use the wave variable into Equation (4) to get the following ODE:
Letting , we get . Then
Solving Equation (25) yields
Setting , we have
Substituting (27) and (28) into Equation (26) gives
Case 1. : For this case, solving Equation (29), we get
Therefore, we obtain the following compacton solutions:
Case 2. : Solving Equation (29), we have the solitary pattern solutions given by
3. The Generalized KdV Equations with Negative Exponents
In fact, Equation (3) and Equation (4) and Equation (5) and Equation (6) have the symmetric property about n respectively. We replace n by −n in Equation (3) and Equation (4) and the corresponding travelling wave solutions in Section 2. So we have the following results:
3.1. Exact Travelling Wave Solutions for Equation (5)
Case 1. : The periodic solutions are given by
Case 2. : The soliton solutions have the forms of
3.2. Exact Travelling Wave Solutions for Equation (6)
Case 1. : In this case, we get the following soliton solutions:
Case 2. : We have the following periodic solutions:
The method based on the reduction of order is a powerful tool for acquiring some special solutions of nonlinear PDEs. In this paper, we study three types of generalized KdV equations with positive and negative exponents by using this mathematical technique. Different from others, this technique carries some partial differential equations into ordinary equations which are easier to be solved. And the analytical expression of travelling wave solutions, containing compactons, solitons, solitary patterns and periodic solutions, are derived.
The obtained results in Section 2 and Section 3 each represent two completely different sets of models, which has been shown that the variation of exponents and coefficient, positive or negative, could cause the quantitative change in the physical structure of the solutions. The physical structures of the compactons solutions and the solitary patterns solutions deepen our understanding of many scientific processes, such as the super deformed nuclei, preformation of cluster in hydrodynamic models, the fission of liquid drops, and the inertial fusion.