Journal of Applied MathematicsVolume 2012 (2012), Article ID 236875, 23 pagesdoi:10.1155/2012/236875

Research Article

Exact Traveling Wave Solutions of Explicit Type, Implicit Type, and Parametric Type for Equation

Xianbin Wu,¹ Weiguo Rui,² and Xiaochun Hong³

¹Junior College, Zhejiang Wanli University, Ningbo 315100, China²College of Mathematics, Honghe University, Mengzi, Yunnan 661100, China³College of Statistics and Mathematics, Yunnan University of Finance and Economics, Kunming, Yunnan 650221, China

Received 9 December 2011; Accepted 22 January 2012

Academic Editor: J. Biazar

Copyright © 2012 Xianbin Wu et al. 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.

Abstract.

By using the integral bifurcation method, we study the nonlinear equation for all possible values of and . Some new exact traveling wave solutions of explicit type, implicit type, and parametric type are obtained. These exact solutions include peculiar compacton solutions, singular periodic wave solutions, compacton-like periodic wave solutions, periodic blowup solutions, smooth soliton solutions, and kink and antikink wave solutions. The great parts of them are different from the results in existing references. In order to show their dynamic profiles intuitively, the solutions of , , , , and equations are chosen to illustrate with the concrete features.

1. Introduction

In this paper, we will investigate some new traveling-wave phenomena of the following nonlinear dispersive equation [1]: where and are integers and is a real parameter. This is a family of fully KdV equations. When , (1.1) as a role of nonlinear dispersion in the formation of patterns in liquid drops was studied by Rosenau and Hyman [1]. In [2–6], the studies show that the model equation (1.1) supports compact solitary structure. In [3], especially Rosenau’s study shows that the branch + (i.e., ) supports compact solitary waves and the branch − (i.e., ) supports motion of kinks, solitons with spikes, cusps or peaks. In [7, 8], Wazwaz developed new solitary wave solutions of (1.1) with compact support and solitary patterns with cusps or infinite slopes under , respectively. In [9], by using the extend decomposition method, Zhu and Lü obtained exact special solutions with solitary patterns for (1.1). In [10], by using homotopy perturbation method (HPM), Domairry et al. studied the (1.1); under particular cases, they obtained some numerical and exact compacton solutions of the nonlinear dispersive and equations with initial conditions. In [11], by variational iteration method, Tian and Yin obtained new solitary solutions for nonlinear dispersive equations ; under particular values of and , they obtained shock-peakon solutions for equation and shock-compacton solutions for equation. In [12], the nonlinear equation is studied by Wazwaz for all possible values of and . In [13], by using Adomian decomposition method, Zhu and Gao obtained new solitary-wave special solutions with compact support for (1.1). In [14], by using a new method which is different from the Adomian decomposition method, Shang studied (1.1) and obtained new exact solitary-wave solutions with compact. In [15, 16], 1-soliton solutions of the equation with generalized evolution are obtained by Biswas. In [17], the bright and dark soliton solutions for equation with -dependent coefficients are obtained by Triki and Wazwaz, especially, when , the equation was studied by many authors; see [18–24] and references cited therein. Defocusing branch, Deng et al. [25] obtained exact solitary and periodic traveling wave solutions of equation. Also, under some particular values of and , many authors considered some particular cases of equation. Ismail and Taha [26] implemented a finite difference method and a finite element method to study two types of equations and . A single compacton as well as the interaction of compactons has been numerically studied. Then, Ismail [27] made an extension to the work in [26], applied a finite difference method on equation, and obtained numerical solutions of equation [28]. Frutos and Lopez-Marcos [29] presented a finite difference method for the numerical integration of equation. Zhou and Tian [30] studied soliton solution of equation. Xu and Tian [31] investigated the peaked wave solutions of equation. Zhou et al. [32] obtained kink-like wave solutions and antikink-like wave solutions of equation. He and Meng [33] obtain some new exact explicit peakon and smooth periodic wave solutions of the equation by the bifurcation method of planar systems and qualitative theory of polynomial differential system.

From the aforementioned references, and references cited therein, it has been shown that (1.1) is a very important physical and engineering model. This is a main reason for us to study it again. In this paper, by using the integral bifurcation method [34–36], we mainly investigate some new exact solutions such as explicit solutions of Jacobian elliptic function type with low-power, implicit solutions of Jacobian elliptic function type, periodic solutions of parametric type, and so forth. We also investigate some new traveling wave phenomena and their dynamic properties.

The rest of this paper is organized as follows. In Section 2, we will derive the equivalent two-dimensional planar system of (1.1) and its first integral. In Section 3, by using the integral bifurcation method, we will obtain some new traveling wave solutions and discuss their dynamic properties; some phenomena of new traveling waves are illustrated with the concrete features.

2. The Equivalent Two-Dimensional Planar System to (1.1) and Its First Integral Equations

We make a transformation with , where the is a nonzero constant as wave velocity. Thus, (1.1) can be reduced to the following ODE: Integrating (2.1) once and setting the integral constant as zero yields Let . The equation (2.2) can be reduced to a 2D planar system: where . Obviously, the solutions of (2.2) include the solutions of (2.3) and constant solution . We notice that the second equation in (2.3) is not continuous when ; that is, the function is not defined by the singular line . Therefore, we make the following transformation: where is a free parameter. Under the transformation (2.4), (2.3), and combine to make one 2D system as follows: Clearly, (2.5) is equivalent to (2.2). It is easy to know that (2.3) and (2.5) have the same first integral as follows: where is an integral constant. From (2.6), we define a function as follows: It is easy to verify that (2.5) satisfies Therefore, (2.5) is a Hamiltonian system and is an integral factor. In fact, (2.7) can be rewritten as the form , where and with . denotes kinetic energy, and denotes potential energy. Especially, when becomes a constant 2. In this case, the kinetic energy only depends on movement velocity of particle; it does not depend on potential function . So, according to Theorem 3.2 in [37], it is easy to know that (2.5) is a stable and nonsingular system when ; in this case its solutions have not singular characters. When , (2.5) becomes a singular system; in this case some solutions of (2.5) have singular characters.

For the equilibrium points of the system (2.5), we have the following conclusion.

Case 1. When is even number, (2.5) has two equilibrium points and . From (2.7), we obtain

Case 2. When is odd number and , (2.5) has three equilibrium points and . From (2.7), we also obtain and Obviously, if is odd, then . If is even, then . Then whether is odd number or even number.

3. Exact Solutions of Explicit Type, Implicit Type, and Parametric Type and Their Properties

3.1. Exact Solutions and Their Properties of (1.1) under

Taking , (2.6) can be reduced to(i) When , (3.1) can be rewritten as Substituting (3.2) into the first expression in (2.5) yields Noticing that equation has two roots , we take as the initial value. Using this initial value, integrating (3.2) yields After completing the aforementioned integral, we solve this equation; thus we obtain Substituting (3.5) into (2.4), then integrating it yields Thus, we respectively obtain a periodic wave solution and solitary wave solution of parametric type for the equation as follows: On the other hand, (3.1) can be rewritten as Using as the initial value, substituting (3.9) into the first expression in (2.3) directly, we obtain an integral equation as follows: Completing the aforementioned integral equation, then solving it, we obtain a periodic solution and a hyperbolic function solution as follows: Obviously, the solution (3.7) is equal to the solution (3.11); also the solution (3.8) is equal to the solution (3.12). Similarly, taking the as initial value, substituting (3.9) into the first expression in (2.3), then integrating them, we obtain another periodic solution and another hyperbolic function solution of equation as follows. In fact, the solutions (3.11) and (3.13) have been appeared in [35], so we do not list similar solutions anymore at here. Next, we discuss a interesting problem as follows.

When , from (3.11) and (3.13), we can construct two compacton solutions as follows: The shape of compacton solutions (3.15) and (3.16) changes gradually as the value of parameter increases. For example, when , respectively, the shapes of compacton solution (3.15) are shown in Figure 1.(ii) When , (3.1) can be directly reduced to Equation (3.17) is a nonsingular equation. Using as initial value and then substituting (3.17) into the first expression in (2.3) directly, we obtain a smooth solitary wave solution and a periodic wave solution of equation as follows: Also, the shape of solitary wave solution (3.18) changes gradually as the value of parameter increases. When , respectively, its shapes of compacton solution (3.18) are shown in Figure 2.(iii)When is even number and , (3.1) can be reduced to It is easy to know that has three roots and with when . In fact, . Using these three roots as initial value, respectively, then substituting (3.20) into the first expression in (2.3), we obtain three integral equations as follows: Completing the previous three integral equations, then solving them, we obtain three periodic solutions of Jacobian elliptic function for equation as follows:

(a)

(b)

(c)

Figure 1: The solution in (3.15) shows a shape of compacton for parameters .

(a)

(b)

(c)

Figure 2: The solution in (3.18) shows a shape of compacton for parameters .

The solutions (3.22) and (3.24) show two shapes of periodic wave with blowup form, which are shown in Figures 3(a) and 3(c). The solution (3.23) shows a shape of periodic cusp wave, which is shown in Figure 3(b).(iv) When , (3.1) can be directly reduced to It is easy to know that the function , where . Using and as initial values, respectively, substituting (3.25) into the first expression in (2.3), we obtain four elliptic integral equations as follows.(1)When ,(2)When ,(3)When ,(4)When , Corresponding to (3.26), (3.27), (3.28), and (3.29), respectively, we obtain four periodic solutions of elliptic function type for equation as follows: where with given previously.

(a) Periodic blowup wave

(b) Periodic cusp wave

(c) Periodic blowup wave

Figure 3: Three periodic waves of solutions (3.22), (3.23), and (3.24) for parameters .

The solution (3.30) shows a shape of periodic wave with blowup form, which is shown in Figure 4(a). The solution (3.31) shows s shape of compacton-like periodic wave, which is shown in Figure 4(b). The profile of solution (3.32) is similar to that of solution (3.30). Also the profile of solution (3.33) is similar to that of solution (3.31). So we omit the graphs of their profiles here.(v) When , (3.1) can be directly reduced to Suppose that is one of roots for equation . Clearly, the 0 is its one root. Anyone solution of equation can be obtained theoretically from the following integral equations: The left integral of (3.35) is called hyperelliptic integral for when the degree is greater than four. Let . Thus, (3.35) can be reduced to In fact, we cannot obtain exact solutions by (3.36) when the degree is grater than five. But we can obtain exact solutions by (3.36) when , and . Under these particular conditions, taking as initial value, (3.36) becomes Let . We obtain and . Thus, (3.37) can be transformed to Completing (3.38) and refunded the variable , we obtain two implicit solutions of elliptic function type for equation as follows: where . The solutions also can be rewritten as where the function is the incomplete Elliptic integral of the first kind.

(a)

(b)

Figure 4: Two different periodic waves on solutions (3.30) and (3.31) for given parameters.

The two solutions in (3.40) are asymptotically stable. Under , as . Under ,   as . The graphs of their profiles are shown in Figure 5.

(a) Asymptotically stable wave

(b) Asymptotically stable wave

Figure 5: Waveforms of two asymptotically stable solutions in (3.40) when .

3.2. Exact Solutions and Their Properties of (1.1) under

In this subsection, under the conditions , we will investigate exact solutions of (1.1) and discuss their properties. When , (2.6) can be reduced to Substituting (3.41) into the first expression of (2.3) yields where is one of roots for equation . However we cannot obtain any exact solutions by (3.42) when the degrees are more great, because we cannot obtain coincidence relationship among different degrees . But, we can always obtain some exact solutions when the degree is not greater than four. For example, by using (3.42) directly, we can also obtain many exact solutions of and equations; see the next computation and discussion.(i) If , then (3.41) can be reduced to Taking as Hamiltonian quantity, substituting (3.43) and into the first expression of (2.5) yields Then has four roots, two real roots, and two complex roots as follows:

with .(1)When and , taking as initial value, then integrating (3.44) yields Solving the aforementioned integral equation yields where and with   and Substituting (3.47) and into (2.4) yields where , the   is an elliptic integral of the third kind, and the function satisfies the following three cases, respectively:

In the previous three cases, . Thus, by using (3.47) and (3.48), we obtain a parametric solution of Jacobian elliptic function for equation as follows:(2)When and , taking as initial value, integrating (3.44) yields Solving the aforementioned integral equation yields where and are given in case (1). Substituting (3.51) and into (2.4) yields where , is an elliptic integral of the third kind, and the function satisfies the following three cases, respectively:

In the previous three cases, . Thus, by using (3.51) and (3.52), we obtain another parametric solution of Jacobian elliptic function for equation as follows: In addition, when ,   has four complex roots; in this case, we cannot obtain any useful results for equation. When , the case is very similar to (3.52); that is, the equation has two real roots and two complex roots. So we omit the discussions for these parts of results.

In order to describe the dynamic properties of the traveling wave solutions (3.49) and (3.53) intuitively, as an example, we draw profile figure of solution (3.53) by using the software Maple, when , see Figure 6(a).

(a) Peculiar compacton wave

(b) Bounded region of independent variable

Figure 6: Peculiar compacton wave and its bounded region of independent variable .

Figure 6(a) shows a shape of peculiar compacton wave; its independent variable is bounded region (i.e., ); see Figure 6(b). From Figure 6(a), we find that its shape is very similar to that of the solitary wave, but it is not solitary wave because when ,   . So, this is a new compacton.(ii) Under , taking as Hamiltonian quantity, (3.42) can be reduced to where is one of roots for the equation . Clearly, this equation has three real roots, one single root and two double roots . If , then the function ; if , then the function . In these two conditions, taking as initial value and completing the (3.54), we obtain a periodic solution and a solitary wave solutions for as follows: Similarly, taking as initial value, we obtain two periodic solutions for as follows:(iii) Under , taking arbitrary constant as Hamiltonian quantity, (3.42) can be reduced to where ,  Write . It is easy to know that as ; this case is same as case (ii). So, we only discuss the case in the next.

When satisfy ,   has three real roots such as with and . Under these conditions, taking the as initial values replacing , respectively, (3.57) can be reduced to the following three integral equations:

Integrating the (3.58), then solving them, respectively, we obtain three periodic solutions of elliptic function type for as follows: where  ,  , and .(iv) When , taking the constant as Hamiltonian quantity, (3.42) can be reduced to Clearly, has two real roots . Taking as initial value, solving (3.62), we obtain a kink wave solution and an antikink wave solution for as follows: where shows that the waves defined by (3.63) are reverse traveling waves.(v) Under , taking arbitrary constant as Hamiltonian quantity and , (3.42) can be reduced to

or Clearly, has four real roots if or ; it has two real roots and two complex roots if or ; it has not any real roots if or .(1)Under the conditions or , taking as an initial value, (3.64) and (3.65) can be reduced to where . Solving the integral equations (3.66), we obtain two periodic solutions of Jacobian elliptic function for equation as follows: where ,

, where . The case for taking as initial values can be similarly discussed; here we omit these discussions because these results are very similar to the solutions (3.67) and (3.68).(2)Under the conditions or , respectively taking as initial value, (3.64) and (3.65) can be reduced to Solving the aforementioned two integral equations, we obtain two periodic solutions of Jacobian elliptic function for equation as follows: where ,  ,   with ,   ,   , and are given previously.

Among these aforementioned solutions, (3.59) shows a shape of solitary wave for given parameters which is shown in Figure 7(a). Equation (3.60) shows a shape of smooth periodic wave for given parameters which is shown in Figure 7(b). Also (3.61) shows a shape of smooth periodic wave for given parameters which is shown in Figure 7(c). Equation (3.63) shows two shapes of kink wave and antikink wave for given parameters which are shown in Figures 7(d)–7(e). Equation (3.68) shows a shape of singular periodic wave for given parameters which is shown in Figure 7(f).

(a) Bright soliton

(b) Smooth periodic wave

(c) Smooth periodic wave

(d) Antikink wave

(e) Kink wave

(f) Singular periodic wave

Figure 7: The graphs of six kinds of waveforms for solutions (3.59), (3.60), (3.61), (3.63), and (3.68).

4. Conclusion

In this work, by using the integral bifurcation method, we study the nonlinear equation for all possible values of and . Some travelling wave solutions such as normal compactons, peculiar compacton, smooth solitary waves, smooth periodic waves, periodic blowup waves, singular periodic waves, compacton-like periodic waves, asymptotically stable waves, and kink and antikink waves are obtained. In order to show their dynamic properties intuitively, the solutions of , , , , and equations are chosen to illustrate with the concrete features; using software Maple, we display their profiles by graphs; see Figures 1–7. These phenomena of traveling waves are different from those in existing literatures and they are very interesting. Although we do not know how they are relevant to the real physical or engineering problem for the moment, these interesting phenomena will attract us to study them further in the future works.

Acknowledgments

The authors thank the reviewers very much for their useful comments and helpful suggestions. This work was financially supported by the Natural Science Foundation of China (Grant no. 11161038). It was also supported by the Natural Science Foundations of Yunnan Province (Grant no. 2011FZ193) and Zhejiang Province (Grant no. Y2111160).

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