Research Article | Open Access
Exact Solutions for a Third-Order KdV Equation with Variable Coefficients and Forcing Term
The general projective Riccati equation method and the Exp-function method are used to construct generalized soliton solutions and periodic solutions to special KdV equation with variable coefficients and forcing term.
It is well known that in an early phase of the development of the solitons theory, there were already many applications in physics and engineering. In particular, traveling waves as solutions of the KdV equation
have been of some interest since 150 years. Some generalizations of this last equation have been studied recently. For instance, the equation
where , are arbitrary constants, which have applications in physics, has been analyzed in  from the point of view of its exact solutions. The search of explicit solutions to nonlinear partial differential equations (NLPDEs) using analytic methods is not an easy task. However, the use of computational methods facilitates this work. Some powerful computational methods such as the tanh method , the generalized tanh method [3, 4], the extended tanh method [5–10], the improved tanh-coth method [11–13], the Exp-function method [14–18], the modified Exp-function method , the Cole-Hopf transformation , the projective Riccati equation method (PREM) [21, 22], the generalized projective Riccati equations method [23–25], the extended hyperbolic function method , and many other methods have been developed in this direction. The PREM and the Exp-function method have been used in a satisfactory form to solve some NLPDEs [15–17, 24, 25, 27–30]. In this paper, we use this last two methods to obtain soliton and periodic solutions to the following special KdV equation with variable coefficients and forcing term:
which results from (1.3) by taking and .
We suppose that the solution to (1.3) has the form
Therefore, (1.3) reduces to
Now we consider the transformation
2. The Exp-Function Method
Recently He and Wu  have introduced the Exp-function method to solve nonlinear equations. In particular, the Exp-function method is a useful tool for solving nonlinear equations with high nonlinearity. The method has been used in a satisfactory way by other authors to solve a great variety of nonlinear wave equations [14–19]. The Exp-function method is very simple and straightforward and is based on a priori assumption that traveling wave solutions to a nonlinear partial differential equation in the form
can be found using the expression
where , , , and are positive integers which could be freely chosen; and are unknown constants to be determined. According to this, we suppose that solutions to (1.8) can be expressed in the form
where , , , and are positive integers which are unknown to be determined later; and are unknown constants. We have the following two cases.
2.1. Case 1: and
In this case, the trial solution to (1.8) becomes
Substituting (2.4) into (1.8) we obtain a polynomial equation in the variable . Equating to zero the coefficients of all powers of yields a set of algebraic equations. Solving it with the aid of a computer, we get , , , and , and, from (1.5), (1.7), and (2.3), one solution to (1.3) is given by
and , , and are arbitrary real or complex numbers.
2.2. Case 2: and
In this case, the trial solution to (1.8) becomes
As in the first case, we obtain an algebraic system. Solving it gives
and , , , and are arbitrary real or complex numbers.
3. General Projective Riccati Equation Method
The projective Riccati equation method was introduced initially in  and generalizations of this method have been used in a satisfactory way by several authors to solve nonlinear partial equations [22–25]. Using this last method [24, 25, 27–30], we seek solutions to (1.8) in the form
where , , are constants and and satisfy the system
In (3.2), and and are certain constants. These equations have following solutions.
Case 1. When and ,
Case 2. When and ,
3.1. Periodic and Soliton Solutions
Periodic and soliton solutions are obtained when and and it corresponds to the first two cases. Substituting (3.1), along with (3.2) and (3.7) into the left hand of (1.8) and collecting all terms with the same power in , we get a polynomial in the variables and . We may choose . Thus, we seek solutions to (1.8) in the form
We equate each coefficient of the polynomial to zero. This will give an overdetermined system of algebraic equations involving the parameters , , and , , and the unknown function . Having determined these parameters, we may determine , and using (1.5) we obtain an exact solution in a closed form. The corresponding system reads.(i)(ii)(iii)(iv)(v), (vi)(vii)(viii)(ix)(x)
In the equations above, , , and . Solving the previous system with the aid of a computer, we obtain many solutions to (1.3). These solutions may be obtained from (1.5) and are given by (3.9)–(3.19). In these formulas, and , are arbitrary parameters. In a formula containing , we suppose that and if the expression appears, we choose . If a formula involves and (see, e.g., (3.17)) we consider that and
Second Group. :.
(v), , , , :(vi), , , , :(vii), , , , :(viii), , , , , :
3.2. Rational Solutions
Note that this last equation has the general solution
We now substitute (3.20) into (1.8), and using the relation we obtain an equation whose left-hand side is a polynomial in the variable . We equate each coefficient of this polynomial to zero and we get the following algebraic system.(i)(ii)(iii)(iv)
Solving this system gives , , and
A rational solution of (1.8) is given by
In this paper, by using the projective Riccati equation and the Exp-function methods, with the help of a symbolic computation engine, we obtain exact solutions for a generalized KdV equation with forcing term (1.3). The methods certainly works well for a large class of very interesting nonlinear equations. The main advantage of these methods is their capability of greatly reducing the size of computational work compared to existing techniques such as the pseudospectral method, the inverse scattering method, Hirota's bilinear method, and the truncated Painlevé expansion. The Exp-function method gives us more general solutions with some free parameters than the projective Riccati equation method. It also has other interesting applications. For instance, the Exp-function method may be applied not only to differential equations but also to differential-difference equations or Stochastic equations [32–35].
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Copyright © 2009 Alvaro H. Salas S and Cesar A. Gómez S. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.