Abstract
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 simplest equation method and its variants. The solutions obtained are general solutions which are in the form of hyperbolic, trigonometric, and rational functions. These methods are more effective and simple than other methods and a number of solutions can be obtained at the same time.
1. Introduction
In recent decades, the study of nonlinear partial differential equations (NLEEs) modelling physical phenomena has become an important research topic. Seeking exact solutions of NLEEs has long been one of the central themes of perpetual interest in mathematics and physics. With the development of symbolic computation packages like Maple and Mathematica, many powerful methods for finding exact solutions have been proposed, such as the homogeneous balance method [1], the extended -expansion method [2], the auxiliary equation method [3], the sine-cosine method [4], the Jacobi elliptic function method [5], the exp-function method [6], the tanh-function method [7], the -expansion method [8], and the -expansion method [9, 10].
The simplest equation method is a very powerful mathematical technique for finding exact solutions of nonlinear ordinary differential equations. It has been developed by Kudryashov [11, 12] and used successfully by many authors for finding exact solutions of ODEs in mathematical physics [12, 13].
Recently, Bilige et al. introduced a method called the extended simplest equation method, as an extension of the simplest equation method, to look for the exact traveling wave solutions of NLEEs [14, 15]. This method can construct different forms of exact traveling wave solutions which cannot be obtained by using the tanh-function method, -expansion method, and the exp-function method.
In 1995, based on the physical and asymptotic considerations, Fokas [16] derived the following generalized KdV equation: which is an important water wave model, where , , , , , , , , and . Regarding the , , , , , , and as free parameters and using the to replace the , (1) becomes the following PDE: which is given by Tzirtzilakis et al. in [17]. They called it high-order wave equation of KdV type. Just as Tzirtzilakis et al. [17] said these two equations are both water wave equations of KdV type, which are more physically and practically meaningful.
Assuming that the waves are unidirectional and neglecting terms of , (1) can be reduced to the classical KdV equation
Assuming that the is less than , this implies ; then and . Neglecting two high-order infinitesimal terms of , (1) can be reduced to another high-order wave equation of KdV type [17–20] as follows:
Equation (4) is a special case of (1) for .
If only we neglect the highest-order term of , then (1) can be reduced to a new generalized KdV equation as follows:
In fact, (5) is another special case of (1) for . It is also third-order approximate equation of KdV type.
Of course, on describing dynamical behaviors of water waves, (4) is only a rough approximative model of (1) compared with (5); that is, the precision of model (5) is better than that of model (4) on describing dynamical behaviors of water waves. In other words, model (5) exhibits a much richer phenomenology than the model (4). Therefore, the investigation of exact traveling wave solutions for (5) is more practically meaningful than that of (4).
Equation (5) is studied by Wu et al. in [21] using the integral bifurcation method and some exact solutions in parameter form are given. In [22], some exact traveling wave solutions of (5) are given by using the extended -expansion method [2]. In this paper, regarding the as free parameters and by using the simplest equation method and its variants, we will investigate exact traveling wave solutions of (5).
The organization of the paper is as follows. In Section 2, a brief description of the simplest equation method and its variants for finding traveling wave solutions of nonlinear equations are given. In Section 3, we will study (5) by the simplest equation methods and its variants. Finally conclusions are given in Section 4.
2. Description of the Simplest Equation Method and Its Variants
Consider a general nonlinear partial differential equation (PDE) for in the form where is a polynomial in its arguments.
By taking and , we look for traveling wave solutions of (6) and transform it to the ordinary differential equation (ODE)
2.1. The Simplest Equation Method
Suppose the solution of (7) can be expressed as a finite series in the form where satisfies the Bernoulli or Riccati equation, is a positive integer that can be determined by balancing procedure, and are parameters to be determined.
The Bernoulli equation we consider in this paper is where and are constants. Its solutions can be written as where , , and are constants.
For the Riccati equation where , , and are constants. Equation (11) has 27 special solutions [23]; in this paper, we will use the following two solutions: where .
Substitute (8) into (7) with (9) (or (11)); then the left-hand side of (7) is converted into a polynomial in ; equating each coefficient of the polynomial to zero yields a set of algebraic equations for . Solving the algebraic equations by symbolic computation, we can determine those parameters explicitly.
Assuming that the constants can be obtained and substituting the results into (8), then we obtain the exact traveling wave solutions for (6).
Remark 1. In (9), when and we obtain the Bernoulli equation
Equation (13) admits the following exact solutions: when , and when .
2.2. The Generalized Simplest Equation Method
We Suppose the solution of (7) can be expressed in the following form: where are arbitrary constants to be determined later and is where satisfies the following auxiliary ordinary differential equation (ODE): where the prime denotes derivative with respect to . , , , and are real parameters.
To determine the positive integer , take the homogeneous balance between the highest-order nonlinear terms and the highest-order derivatives appearing in (7). Substituting (16) and (18) including (17) into (7) with the value of obtained and we obtain polynomials in and . Then, we collect each coefficient of the resulted polynomials to zero, yields a set of algebraic equations for , , and .
Suppose that the value of the constants , , and can be found by solving the algebraic equations which are obtained. Since the general solution of (18) is well known to us, substituting the values of , , and into (16), we can obtain a more general type and new exact traveling wave solutions of the nonlinear partial differential equation (6).
The general solutions of (18) can be listed as follows:(1)when and , (2)when and , (3)when and ,
2.3. The Extended Simplest Equation Method
We Suppose the solution of (7) can be expressed in the following form: where are constants and . The positive number can be determined by considering the homogeneous balance between the highest-order derivatives and nonlinear terms appearing in (7). The function satisfies the second order linear ODE in the form where and are constants. Equation (23) has three types of general solution with double arbitrary parameters as follows: where and are arbitrary constants.
By substituting (22) into (7) and using the second order linear ODE (23) and (25), collecting all terms with the same order of and together, the left-hand side of (7) is converted into another polynomial in and . Equating each coefficient of these different power terms to zero yields a set of algebraic equations for , and .
Assume constants , , and can be determined by solving the nonlinear algebraic equations. Then substituting these terms and the general solutions (24) of (23) into (7), we can obtain more exact traveling wave solutions of (6).
3. Exact Solutions of (5)
Making a transformation , with , (5) can be reduced to the following ODE: where is wave velocity which moves along the direction of -axis and . Integrating (11) once and setting the integral constant as yield
3.1. Using the Simplest Equation Method
3.1.1. Solutions of (27) Using the Bernoulli Equation as the Simplest Equation
Considering the homogeneous balance between and , we get , so the solution of (27) is in the form
Substituting (28) into (27) and making use of the Bernoulli equation (9) and then equating the coefficients of the functions to zero, we obtain an algebraic system of equations in terms of , , , and as follows:, + + + + + + , + + + + + − , + + + + + .
On solving the above algebraic equations using the Maple, we get the following results:
Therefore, using solutions (10) of (9), ansatz (28), we obtain the following exact solution of (5): where , and are determined in (29), and , , , , , , and are arbitrary constants.
3.1.2. Solutions of (27) Using Riccati Equation as the Simplest Equation
Suppose the solution of (27) is of the form
Substituting (32) into (27) and making use of the Riccati equation (11) and then equating the coefficients of the functions to zero, we obtain an algebraic system of equations in terms of , , , and as follows:, , + + + + + + + + , + + + + + + + + − , + + + + + + + − + = .
On solving the above algebraic equations using the Maple, we get the following results: where .
Therefore, using solutions (12) of (11), ansatz (32), we obtain the following exact solution of (5): where , , , and are determined in (33), and , , , , , , and are arbitrary constants.
3.2. Using Generalized Simplest Equation
Suppose the solution of (27) is of the form where , , and are constants to be determined later and function satisfies (17) and auxiliary differential equation (18).
Substituting (36) together with (17) and (18) into (27), the left-hand side is converted into polynomials in , . We collect each coefficient of these resulted polynomials to zero, yields a set of simultaneous algebraic equations for , , , , and . Solving this system of algebraic equations, with the aid of Maple, we obtain where and . , , , , , , , and are free parameters.
Substituting (37) into (36), along with (19) and simplifying, yields the following traveling wave solutions (if , but ; , but ), respectively: where , , , , , , and are determined in (37), and , , , , , , and , are arbitrary constants.
Substituting (37) into (36), along with (20) and simplifying, yields the following traveling wave solutions (if , but ; , but ), respectively: where , , , , , and are determined in (37), and , , , , , , and , are arbitrary constants.
Substituting (37) into (36), along with (21), and simplifying, our obtained solution becomes where , , and are determined in (37), and , , , , , , and are arbitrary constants.
3.3. Using Extended Simplest Equation
Suppose the solution of (27) is of the form where , , and are constants to be determined later and function satisfies the second order linear ODE (23).
By substituting (42) into (27) and using the second order linear ODE expressions (23) and (25), collecting all terms with the same order of and together, the left-hand side of (27) is converted into another polynomial in and . Equating each coefficient of these different power terms to zero yields a set of algebraic equations for , , , , , , , , , , , and . Solving these equations, we obtain the following results.
If , we obtain
Substituting (43)–(52) into (42) and making use of solutions (24) of (23), we can obtain, respectively, exact traveling wave solutions expressed by hyperbolic functions of (5). Consider where , , , and are determined in (43), and , , , , , , , and are arbitrary constants. Consider where , , , and are determined in (44), and , , , , , , and are arbitrary constants. Consider where , , , , and are determined in (45), and , , , , , and are arbitrary constants. Consider where , , , , and are determined in (46), and , , , , , and are arbitrary constants. Consider where , are determined in (47), and , , , , , , and are arbitrary constants. Consider where , , , , and are determined in (48), and , , , , and are arbitrary constants. Consider where , , , , and are determined in (49), and , , , , , and are arbitrary constants. Consider where , , , and are determined in (50), and , , , , , , and are arbitrary constants. Consider where , , , and are determined in (51), and , , , , , , and are arbitrary constants. Consider where , , , , and are determined in (52), and , , , , , , and are arbitrary constants.
If , we obtain
Substituting (63)–(69) into (42) and making use of solutions (24) of (23), we can obtain, respectively, exact traveling wave solutions expressed by trigonometric functions of (5). Consider where , , , and are determined in (63), and , , , , , , and are arbitrary constants. Consider where , , , , and are determined in (64), and , , , , , and are arbitrary constants. Consider where , , , and are determined in (65), and , , , , , , and are arbitrary constants. Consider where , , , , and are determined in (66), and , , , , and are arbitrary constants. Consider where , , , and are determined in (67), and , , , , , , and are arbitrary constants. Consider where , , , and are determined in (68), and , , , , , , and are arbitrary constants. Consider where , , , , and are determined in (69), and , , , , , , and are arbitrary constants.
If , we obtain
Substituting (77)–(82) into (42) and making use of solutions (24) of (23), we can obtain, respectively, exact traveling wave solutions expressed by rational functions of (5). Consider where , , , and are determined in (77), and , , , , , , and are arbitrary constants. Consider where , , , and are determined in (78), and , and are arbitrary constants.Consider where , , , and are determined in (79), and , , , , , , and are arbitrary constants. Consider where , , , and are determined in (80), and , , , , , , and are arbitrary constants. Consider where , , , and are determined in (81), and , , , , , , and are arbitrary constants. Consider where , , , and are determined in (82), and , , , , , , , and are arbitrary constants.
4. Figures of Some Exact Solutions
In this section, some typical wave figures are given as in Figures 1, 2, 3, 4, and 5.
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| (a) , , |
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5. Conclusions
In this paper, with the aid of Maple, we successfully obtained wider classes of exact traveling solutions of (5) by using the simplest equation method and its variants. And some new exact solutions expressed by hyperbolic function, trigonometric function, and rational functions are obtained and some typical wave figures are given including periodic wave, solitary wave, kink wave, and some new types. The simplest equation method and its variants can construct different forms of exact traveling wave solutions which cannot be obtained by using the tanh-function method, -expansion method, -expansion method, and the exp-function method. The correctness of all the solutions is verified by substituting them into original equation (5). Comparing with [21, 22], it is easy to see that our method is more straightforward and the form of the solutions obtained in our paper is also more simple and many solutions are new. The related results are enriched.
Conflict of Interests
The author declares that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
This work was supported by the National Natural Science Foundation of China (11161020; 11361023), the Natural Science Foundation of Yunnan Province (2011FZ193), and the Natural Science Foundation of Education Committee of Yunnan Province (2012Y452 and 2013C079).