Journal of Applied Mathematics

Volume 2013 (2013), Article ID 183159, 14 pages

http://dx.doi.org/10.1155/2013/183159

Research Article

Bifurcations and Parametric Representations of Peakon, Solitary Wave and Smooth Periodic Wave Solutions for a Two-Component Degasperis-Procesi Equation

Bin He,¹ Qing Meng,² and Jinhua Zhang¹

¹College of Mathematics, Honghe University, Mengzi, Yunnan 661100, China

²Department of Physics, Honghe University, Mengzi, Yunnan 661100, China

Received 27 May 2013; Revised 26 July 2013; Accepted 26 July 2013

Academic Editor: Jingxin Zhang

Copyright © 2013 Bin He 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 bifurcation method of dynamical systems and the method of phase portraits analysis, we consider a two-component Degasperis-Procesi equation: the existence of the peakon, solitary wave and smooth periodic wave is proved, and exact parametric representations of above travelling wave solutions are obtained in different parameter regions.

1. Introduction

Based on the deformation of bi-Hamiltonian structure of the hydrodynamic type, Chen et al. [1] obtained a two-component Camassa-Holm equation:

where and . This system is integrable in the sense that it has Lax pair and it is a generalization form of the Camassa-Holm equation [2]. A good fact is that it has the peakon and multikink solutions [3].

A two-component generalization of the Degasperis-Procesi equation [4, 5] is read as follows:

where , is a real parameter. When , (2) is reduced to the Degasperis-Procesi equation [6]. Jin and Guo [4] analyze some aspects of blowup mechanism, traveling wave solutions and the persistence properties for (2). The self-similar solutions of (2) have been obtained by yuen [5].

In the present paper, we will investigate the bifurcation set of (2) using the bifurcation theory and the method of phase portraits analysis [7–9] and obtain some exact peakon, solitary wave and smooth periodic wave solutions.

Let , where is the wave speed. By using the travelling wave transformation

we reduce (2) to the following ordinary differential equations:

where is the derivative with respect to .

Suppose that

where , are constants to be determined later.

Substituting (5) into the second equation of (4), we have

Integrating (6) once with respect to and setting the integral constant to zero yield

From (7), we obtain

Solving (8), we find the following set of solutions:

Substituting (9) into (5), we have

Substituting (10) into the first equation of (4), integrating once with respect to , and setting the integral constant to yield

Letting , we get the following planar system:

System (12) is a two-parameter planar dynamical system depending on the parameter set . Since the phase orbits defined by the vector field of system (12) determine all travelling wave solutions of (11), we should investigate the bifurcations of phase portraits of system (12) in -phase plane as the parameters are changed.

Clearly, on such straight line in the phase plane , system (12) is discontinuous. Such system is called a singular travelling wave system by one of authors in [10].

Definition 1 (see [7, 10]). Suppose that is a solution of system (12) for . (i) is called peakon solution if is smooth locally on either side of and and , . (ii) is called a solitary wave solution if . Usually, a peakon solution of (2) corresponds to two heteroclinic orbits of system (12) and a solitary wave solution of (2) corresponds to a homoclinic orbit of system (12). Similarly, a periodic orbit of system (12) corresponds to a smooth periodic wave solution of (2).

Thus, to investigate peakons, solitary waves and smooth periodic waves of (2), we should find all heteroclinic, homoclinic, and periodic orbits of system (12) depending on the parameter space of this system.

The rest of this paper is organized as follows. In Section 2, we discuss the bifurcations of phase portraits of system (12), where explicit parametric conditions will be derived. In Section 3, we give some exact parametric representations of peakon, solitary wave and smooth periodic wave solutions of (2). A short conclusion will be given in Section 4.

2. Bifurcation Sets and Phase Portraits of System (12)

Using the transformation , it carries (12) into the Hamiltonian system

Since both systems (12) and (13) have the same first integral

then two systems above have the same topological phase portraits except for the line . Therefore, we can obtain the bifurcation phase portraits of system (12) from that of system (13). We consider the equilibrium points and their properties for system (13) as follows.

When , it is easy to see that system (13) has two equilibrium points on -axis and two equilibrium points on the line .

When , let

Obviously, is an equilibrium point of system (13) and is a equilibrium point of system (13) if and only if when . Denoting that , , we draw the graph of for given as Figure 1.

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

Figure 1: The graphics of the function when . (a) ,  . (b) , . (c) , . (d): , . (e) , . (f) , . (g) , . (h) , .

From Figure 1 and paying attention to that is always an equilibrium point of system (13), we have the following.

(i)If , then system (13) has two equilibrium points on -axis.(ii)If , then system (13) has four equilibrium points on -axis.(iii)If , then system (13) has three equilibrium points on -axis.(iv)If , then system (13) has two equilibrium points on -axis.

On the other hand, it is also easy to see that if , then system (13) has two equilibrium points on the line and if   , then system (13) has one equilibrium point on the line .

Let be the coefficient matrix of the linearized system of the system (13) at an equilibrium point . Then, we have

and at this equilibrium point, we have

By the theory of planar dynamical systems, we know that for an equilibrium point of a planar integrable system if , then the equilibrium point is a saddle point. If and , then it is a center point. If and the Poincaré index of the equilibrium point is zero, then it is a cusp.

For a fixed , the level curve defined by (14) determines a set of invariant curves of system (13) which contains different branches of curves. As is varied, it defines different families of orbits of (13) with different dynamical behaviors.

Using the property of the equilibrium points and bifurcation theory, we obtain the following results.

(i)When , there is one bifurcation line which divides the region into 2 subregions: , .(ii)When , there are five bifurcation curves as follows:

which divide the -parameter plane into 16 subregions as follows:

where , .

The bifurcation sets and phase portraits of system (13) are shown in Figures 2 and 3.

(a)

(b)

Figure 2: The bifurcation sets and phase portraits of system (13) when . (a) . (b) .

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

(i)

(j)

(k)

(l)

(m)

(n)

(o)

(p)

Figure 3: The bifurcation sets and phase portraits of system (13) when . (a) . (b) . (c) . (d) . (e) (f) . (g) . (h) . (i) . (j) . (k) . (l) . (m) . (n) . (o) . (p) .

3. Exact Parametric Representations of Peakon, Solitary Wave and Smooth Periodic Wave Solutions of (2)

In this section, we present all possible exact parametric representations of peakon, solitary wave and smooth periodic wave solutions through some special phase orbits. Next, we always suppose that , + , , , , and .

3.1. Exact Parametric Representations of Peakon Solutions

(i)   For the given in Figure 2(a), the level curve is shown in Figure 4(a). From Figure 4(a), we see that there are two heteroclinic orbits connecting with saddle points and of system (13) when . Their expressions are

(a) ,

(b) ,

(c) ,

(d) ,

(e) ,

(f) ,

(g) ,

(h) ,

Figure 4: The level curves of system (13) defined by .

Substituting (20) into the and integrating it along the heteroclinic orbits yield

Completing the above integral and solving the equation for , it follows that

Noting that , we get the parametric representation of peakon solution as follows:

(ii)   For the given in Figure 2(b), the level curve is shown in Figure 4(b). From Figure 4(b), we see that there are two heteroclinic orbits connecting with saddle points and of system (13) when . Their expressions are

Substituting (24) into the and integrating it along the heteroclinic orbits yield

Completing the above integral, we can get the parametric representation of peakon solution which is the same as (23).

3.2. Exact Parametric Representations of Solitary Wave Solutions

(i) For the given in Figure 3(d), the level curve is shown in Figure 4(c). From Figure 4(c), we see that there is a homoclinic orbit connecting with a saddle point of system (13) and passing point , and at the same time, there is a homoclinic orbit connecting with the saddle point and passing point when , and their expressions are

respectively, where , , , and are four real roots of

which are obtained by the Cardan formula. For example, , , , when , ; , , , when , ; and , , , when , .

Substituting (26) into the and integrating it along the homoclinic orbit yield

Completing the above integral, we can get the parametric representation of solitary wave solution as follows:

is confirmed by

where , , , , , is Legendre's incomplete elliptic integral of the third kind, is the Jacobian elliptic function (see [11]), and is a new variable.

Substituting (27) into the and integrating it along the homoclinic orbit yield

Completing the above integral, we can get the parametric representation of solitary wave solution as follows:

is confirmed by

where , , , , and .

(ii) For the given in Figure 3(e), the level curve is shown in Figure 4(d). From Figure 4(d), we see that there is a homoclinic orbit connecting with a cusp of system (13) and passing point when , and its expression is

where , , and are three real roots of

which are obtained by the Cardan formula. For example, , , when ; , , when ; and , , when .

Substituting (35) into the and integrating it along the homoclinic orbit yield

Completing the above integral, we can get the parametric representation of solitary wave solution as follows:

is confirmed by

where , , , and .

(iii) For the given in Figure 3(g), the level curve is shown in Figure 4(e). From Figure 4(e), we see that there is a homoclinic orbit connecting with a cusp of system (13) and passing point when , and its expression is

where , , and are three real roots of

which are obtained by the Cardan formula, and , , and . For example, , , when ; , , when ; and , , when .

Substituting (40) into the and integrating it along the homoclinic orbit yield

Completing the above integral, we can get the parametric representation of solitary wave solution as follows:

is confirmed by

where , , , and .

(iv) For the given in Figure 3(l), the level curve is shown in Figure 4(f). From Figure 4(f), we see that there is a homoclinic orbit connecting with a saddle point of system (13) and passing point , and at the same time, there is a homoclinic orbit connecting with the saddle point and passing point when , and their expressions are

respectively, where , , , and are four real roots of (28).

Substituting (45) into the and integrating it along the homoclinic orbit yield

Completing the above integral, we can get the parametric representation of solitary wave solution as follows:

is confirmed by

where , , , , and .

Substituting (46) into the and integrating it along the homoclinic orbit yield

Completing the above integral, we can get the parametric representation of solitary wave solution as follows:

is confirmed by

where , , , and .

(v) For the given in Figure 3(m), the level curve is shown in Figure 4(g). From Figure 4(g), we see that there is a homoclinic orbit connecting with a cusp of system (13) and passing point when , and its expression is

where , , and are three real roots of (36).

Substituting (53) into the and integrating it along the homoclinic orbit yield

Completing the above integral, we can get the parametric representation of solitary wave solution as follows:

is confirmed by

where , , , and .

(vi) For the given in Figure 3(o), the level curve is shown in Figure 4(h). From Figure 4(h), we see that there is a homoclinic orbit connecting with a cusp of system (13) and passing point when , and its expression is

where , , and are three real roots of (41).

Substituting (57) into the and integrating it along the homoclinic orbit yield

Completing the above integral, we can get the parametric representation of solitary wave solution as follows:

is confirmed by

where , , , and .

3.3. Exact Parametric Representations of Smooth Periodic Wave Solutions

(i) For the given in Figure 3(d), the level curve is shown in Figure 4(c). From Figure 4(c), we see that there is one periodic orbit passing points and when , and its expression is

where , , , and are four real roots of (28).

Substituting (61) into the and integrating it along the periodic orbit yield

Completing the above integral, we can get the parametric representation of smooth periodic wave solution as follows:

is confirmed by

where , , , , and .

(ii) For the given in Figure 3(e), the level curve is shown in Figure 4(d). From Figure 4(d), we see that there is one periodic orbit passing points and when , and its expression is

where , , and are three real roots of (36).

Substituting (65) into the and integrating it along the periodic orbit yield

Completing the above integral, we can get the parametric representation of smooth periodic wave solution as follows:

is confirmed by

where , , , , and and are Legendre's incomplete elliptic integrals of the first and second kinds, respectively (see [11]).

(iii) For the given in Figure 3(g), the level curve is shown in Figure 4(e). From Figure 4(e), we see that there is one periodic orbit passing points and when , and its expression is

where , , and are three real roots of (41).

Substituting (69) into the and integrating it along the periodic orbit yield

Completing the above integral, we can get the parametric representation of smooth periodic wave solution as follows:

is confirmed by

where , , , and .

(iv) For the given in Figure 3(l), the level curve is shown in Figure 4(f). From Figure 4(f), we see that there is one periodic orbit passing points and when , and its expression is

where , , , and are four real roots of (28).

Substituting (73) into the and integrating it along the periodic orbit yield

Completing the above integral, we can get the parametric representation of smooth periodic wave solution which is the same as (33).

(v) For the given in Figure 3(m), the level curve is shown in Figure 4(g). From Figure 4(g), we see that there is one periodic orbit passing points and when , and its expression is

where , , and are three real roots of (36).

Substituting (75) into the and integrating it along the periodic orbit yield

Completing the above integral, we can get the parametric representation of smooth periodic wave solution as follows:

is confirmed by

where , , , and .

(vi) For the given in Figure 3(o), the level curve is shown in Figure 4(h). From Figure 4(h), we see that there is one periodic orbit passing points and when , and its expression is

where , , and are three real roots of (41).

Substituting (79) into the and integrating it along the periodic orbit yield

Completing the above integral, we can get the parametric representation of smooth periodic wave solution as follows:

is confirmed by

where , , , and .

Remark 2. From Figures 3(f) and 3(n), we see that (2) has two solitary wave solutions when and two solitary wave solutions when , but the exact parametric representations are very complex, so here we omit them.

4. Conclusion

In this paper, using the bifurcation theory and the method of phase portraits analysis, we investigated the bifurcations of (2) (which is called a two-component Degasperis-Procesi equation) and obtained some exact parametric representations of peakon, solitary wave and smooth periodic wave solutions (see (23), (30), and (63), etc.). We can say that we obtained some new results of (2) in this paper.

Acknowledgments

The authors thank the referees very much for their perceptive comments and suggestions. This work is supported by the Natural Science Foundation of Yunnan Province, China (no. 2013FZ117) and the National Natural Science Foundation of China (no. 11161020).

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