Exact Solitary Wave and Periodic Wave Solutions of a Class of Higher-Order Nonlinear Wave Equations

We study the exact traveling wave solutions of a general fifth-order nonlinear wave equation and a generalized sixth-order KdV equation. We find the solvable lower-order subequations of a general related fourth-order ordinary differential equation involving only even order derivatives and polynomial functions of the dependent variable. It is shown that the exact solitary wave and periodic wave solutions of some high-order nonlinear wave equations can be obtained easily by using this algorithm. As examples, we derive some solitary wave and periodic wave solutions of the Lax equation, the Ito equation, and a general sixth-order KdV equation.

We investigate the traveling wave solutions of the nonlinear wave equation (1) in the form (, ) = ( − ) = (), where  is the wave speed and  =  − .Under the traveling wave coordinates, the nonlinear wave equation (1) can be reduced to a nonlinear ordinary differential equation (ODE) of the independent variable .Integrating the reduced ODE once with respect to  gives  4   4 + where  is a constant of integration.Clearly, () = ( − ) is a traveling wave solution of (1) if and only if () satisfies 2 Mathematical Problems in Engineering (3) with the wave speed  and any constant .The equilibrium points and linearized system of (3) were studied in [7,14,15].By using the method of dynamical systems and Cosgrove's results [16], Li [7,14,15] obtained some exact solitary wave and quasiperiodic wave solutions of the CDG equation.Generally, we have to study the dynamical behavior of the fourth-order ODE (3) in the 4-dimensional phase space, for which it is usually very difficult to obtain the orbits.However, for the case when the first integrals of this equation can be found, this problem can be reduced to the one in the lower dimensional space which might be easier to handle.It has been a successful idea to find exact solutions of nonlinear PDEs by reducing them into ODEs, especially for some solvable ODEs.A lot of methods in the literatures use this idea, for instance, the tanh-function method and extended tanh-function method [3,4,17], simple transformation method [18], the Riccati equation method [19], the Jacobi elliptic function method [20], the expfunction method [21], the homogeneous balance method [22], the (  /)-expansion method [23], and subequation method [24].Also a direct and systematical approach to find exact solutions of nonlinear equations was proposed by using rational function transformations and thus was named as the transformed rational function method by Ma [25].However, if the dynamics and bifurcation of these ODEs are not investigated, some solutions obtained by different methods, which are presented in different forms, can easily be misunderstood as different solutions [26,27].For example, a solution in the form (tanh 2 ) obtained by the tanh-method can be expressed as (1−sech 2 ) or ((  − − ) 2 /(  + − ) 2 ), which also can be derived by the sech-method and the Expfunction method.
Notice that (3) contains the terms  4 / 4 ,  2 / 2 , (/) 2 , and polynomial of .Suppose that  satisfies the equation where   () is a polynomial function of degree ; then which are both polynomials of .This observation motivates us to try to find some possible integer  and undetermined coefficients of the polynomial   such that  solves higherorder equation (3) if  is a solution of (4) which is obviously easier to study.Following the idea we mentioned above, in Section 2, we derive the subequation of a more general fourth-order ODE firstly and then investigate the bifurcations and bounded solutions of (7) through the obtained subequations.By the formulas presented in Section 2 and with the help of computer algebra and symbolic computation, we study the bounded traveling wave solutions of the Lax equation and the Ito equation as examples in Section 3. In Section 4, we extend the formulas obtained in Section 2 to study the exact traveling wave solutions of a generalized sixth-order KdV equation.

Subequations and Exact Solutions of the Fourth-Order Equation (7)
Suppose that  satisfies (4); then the first three terms of the left-hand side of ( 7) are all polynomials of  and their degrees are 2 − 3,  and , respectively.Consequently, we assume  = 3 ( 3 ̸ = 0) to find the possible polynomial  3 () such that it solves (7) if  solves (4).Definition 1.We say that equation A is a subequation of equation B if any solutions of equation A are also solutions of equation B.
We now prove that 4th-order ODE (7) possesses a class of lower-order solvable subequations.
Remark 3. Note that all the denominators in ( 9) are supposed to be nonzero.If some of them are zero, we have to go back to the algebraic equations ( 10) to find the possible solutions.
According to the conclusion of Theorem 2, we know that the fourth-order ODE ( 7) can be reduced into the first-order nonlinear ODE (8) provided that  ≤ (3 + 2) 2 /120, which is really an inspiring result because the bifurcation and the exact solutions of the first-order nonlinear ODE ( 8) have been presented in [8].We recall the theorem below to derive the exact solutions of (7) and thus obtain the exact traveling wave solutions of the fifth-order nonlinear wave equation ( 1) and the sixth-order KdV equation (2).
then the following conclusions hold.
(1) For  0 = 2ℎ + , ( 8) has a bounded solution approaching  +  as  goes to infinity given by a constant solution and an unbounded solution where  0 is an arbitrary constant.
Case (a).If  3 > 0, then, for any is a family of smooth periodic solutions of (8).Here  + = , and sn represents the Jacobian elliptic sineamplitude function.
We now study the Lax equation and the Ito equation as examples of the application of the approach we proposed in this paper.

Exact Traveling Wave Solutions of the Lax Equation.
The Lax equation which has been studied in [3,7] is given by This is (1) with  = 10,  = 20, and  = 30.
Theorem 7. The Lax equation (20) has the following four families of bounded traveling wave solutions.
(1) The Lax equation ( 20) has a peak-form solitary wave solution (see Figure 1) where the wave speed  > 0.
Remark 8. We note that we recover the solutions of the Lax equation obtained in [3] as special cases of our solutions.The solitary wave solutions ( 22) are consistent with the solutions (138) in [3], but the solution ( 21) and other two families of periodic wave solutions are new.

Exact Traveling Wave Solutions of the Ito Equation. The Ito equation [13] is given by
and can be retrieved from (1) by letting  = 3,  = 6, and  = 2.
Theorem 9.The Ito equation ( 28) has the following four families of bounded traveling wave solutions.
(1) It has a peak-form solitary wave solution where the wave speed  > 0.
(2) For any arbitrary constant  0 , the Ito equation ( 28) has a family of peak-form solitary wave solutions The wave speed of this family of waves is zero; that is, these are standing waves.

Application to the General Sixth-Order KdV Equation (2)
We now generalize this approach to study a new sixth-order nonlinear wave equation ( 2) which was derived by Karasu-Kalkanli et al. [6] in 2008.Letting (, ) = ( − V) = (), setting  = /, and integrating it once with respect to , we have where  is an integration constant.
Note that here  0 can be arbitrary number because  is an arbitrary integration constant.From Lemma 11 and Theorem 4, it is easy to see that the traveling wave solutions of the six-order KdV equations (2) can be obtained by detecting the relationship between the coefficients   ,  = 0, 1, 2, 3, which are determined by (36).The two families of exact solutions exist if  1 ,  2 , and  3 defined by (36) satisfy  2  2 − 3 1  3 > 0, because  0 is an integration constant.With the help of Maple, we can obtain the traveling waves of the sixorder KdV equation (2) automatically.
Note that  is an arbitrary integration constant.Consequently, from Theorem 4, we obtain the exact solutions of (38) as (, ) = ∫ (), where  =  − V and () is defined as follows.

Conclusion and Discussion
The exact traveling wave solutions of a general fifth-order nonlinear wave equation and a generalized sixth-order KdV equation were studied in this paper.A systematic algorithm was proposed to study the exact solutions of an associated fourth-order ODE possessing even order derivatives.By using this algorithm, with the help of symbolic computation, the exact solitary wave and periodic wave solutions of a very general class of higher-order nonlinear wave equations can be obtained systematically.Here we obtained two families of solitary wave and periodic wave solutions of the Lax equation, the Ito equation, and a generalized sixth-order KdV equation.
From the results of this paper, it is easy to see that this method can be used to find the traveling wave solutions of higherorder wave equations which can be reduced to the ODEs in the form ( (2) ,  (2−2) , . . .,  (2) ,  2 , ) = 0, where  is a polynomial function.
It has been shown that the equations with nonlinear dispersion, usually related to singular dynamical systems, possess nonsmooth singular wave solutions, such as compacton and peakon, by using the dynamical system method and other methods [14,[29][30][31][32].However, as far as we know, the dynamical system method has been well used to investigate the nonlinear wave equations which can be reduced to planar dynamical systems.Can we find the subequations of the higher-order wave equations with nonlinear dispersion in the form (/) 2 = ()?Here () is a rational function.How the theorems of planar singular dynamical systems [14,33,34] can be applied to find the singular wave solutions to higherorder nonlinear wave equations will be considered in our future work.

Figure 1 :
Figure 1: Three-dimensional portrait of the solitary wave solution of the Lax equation (20) with  = 1.

Figure 2 :
Figure 2: Portrait of the periodic wave solution of the Lax equation (20) with  = 1.(a) Three-dimensional portrait; (b) overhead view with contour plot.
Remark 13.Obviously, this algorithm also can be used to study the sixth-order KdV equation (2) with other coefficients values.