Mathematical Problems in Engineering

Volume 2013 (2013), Article ID 602432, 19 pages

http://dx.doi.org/10.1155/2013/602432

Research Article

Exact Peakon, Compacton, Solitary Wave, and Periodic Wave Solutions for a Generalized KdV Equation

Qing Meng¹ and Bin He²

¹Department of Physics, Honghe University, Mengzi, Yunnan 661100, China

²College of Mathematics, Honghe University, Mengzi, Yunnan 661100, China

Received 29 August 2013; Accepted 25 September 2013

Academic Editor: Jun-Juh Yan

Copyright © 2013 Qing Meng and Bin He. 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.

We employ the approaches of both dynamical system and numerical simulation to investigate a generalized KdV equation, which is presented by Yin (2012). Some peakon, compacton, solitary wave, smooth periodic wave, and periodic cusp wave solutions are obtained, and the planar graphs of the compactons and the periodic cusp waves are simulated.

1. Introduction

To study the role of nonlinear dispersion in the formation of patterns in the liquid drop, Rosenau and Hyman [1] showed in a particular generalization of the KdV equation

which is called equation. They found some solitary waves with compact support in it, which they called compactons. These compactons had the property that the width was independent of the amplitude. Equation (1) has been studied successfully by some authors [2–8]. However, (1) does not exhibit the usual energy conservation law. Instead of (1), Cooper et al. considered the corresponding generalized KdV equation [9]

which can be derived from a Lagrangian. Equation (2) has the same terms as (1), except the relative weights of the terms. Equation (2) also admits compacton solutions. The stability of the compacton solutions to (2) was studied in [10]. In the presence of a linear dispersion term, (2) turns into a generalized KdV equation with combined dispersion [11]

Obviously, for , (3) turns into (2). Yin [11] indicated that (3) has two conservative laws and showed that the smooth solitary waves are stable for any speed of wave propagation.

In this paper, we investigated (3) using the approaches of dynamical system and numerical simulation [12–14], we will give some new exact travelling wave solutions and simulate the compactons and the periodic cusp waves.

Next, we always suppose that . Using the following independent variable transformation:

where is the wave speed, and substituting (4) into (3), we obtain

where “” is the derivative with respect to .

Integrating (5) once with respect to , we have

where is the integral constant.

Letting , we get the following planar dynamical system:

The rest of this paper is organized as follows. In Section 2, we discuss the bifurcation sets and phase portraits of system (7), where explicit parametric conditions will be derived. In Section 3, we give some exact travelling wave solutions which include peakon, compacton, solitary wave, smooth periodic wave, and periodic cusp wave solutions of (3). In Section 4, the numerical simulations of the compactons and the periodic cusp waves are given. A short conclusion will be given in Section 5.

2. Bifurcation Sets and Phase Portraits of System (7)

Using the transformation , it carries (7) into the Hamiltonian system

System (8) has the following first integral:

For a fixed , the level curve defined by (9) determines a set of invariant curves of system (8) which contains different branches of curves. As is varied, it defines different families of orbits of (8) with different dynamical behaviors.

Write , . Clearly, when , system (8) has two equilibrium points at and in -axis, where . When , system (8) has only one equilibrium point at in -axis. When , there exist two equilibrium points of system (8) in line at ), where . When , there is no equilibrium point of system (8) in line .

Let be the coefficient matrix of the linearized system of (8) at equilibrium point ; then we have Trace  and

For an equilibrium point of a planar integrable system, we know that is a saddle point if , a center point if and Trace , and a cusp if and the Poincaré index of is zero.

Since both systems (7) and (8) have the same first integral (9), then two systems above have the same topological phase portraits except the line . Therefore we can obtain the bifurcation sets and phase portraits of system (7) from those of system (8).

By using the properties of equilibrium points and bifurcation method of dynamical systems, we can show that bifurcation sets and phase portraits of system (7) are drawn in Figures 1 and 2.

(a)  

(b)

(c)

(d)

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(i)

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Figure 1: Bifurcation sets and phase portraits of system (7) when .

(a)

(b)

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(d)

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(i)

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Figure 2: Bifurcation sets and phase portraits of system (7) when .

3. Exact Travelling Wave Solutions of (3)

Denote that and ; we present some exact travelling wave solutions of (3) as follows.

3.1. Peakon Solutions

From Figure 2(d), we see that there are two heteroclinic orbits of system (7) defined by connecting with the saddle points and when , , and . Their expressions are

Substituting (11) into and integrating it along the heteroclinic orbits yield equation

Completing (12) and using transformation (4), we can get a peakon solution of (3) as follow:

where . The profiles of (13) are shown in Figures 3(a) and 3(b).

(a)

(b)

(c)

(d)

Figure 3: Peakons of (3) for .

From Figure 2(k), we see that there are two heteroclinic orbits of system (7) defined by connecting with the saddle points and when , , and . Their expressions are

Substituting (14) into and integrating it along the heteroclinic orbits yield equation

Completing (15) and using transformation (4), we can get a peakon solution of (3) which is the same as (13). The profiles are shown in Figures 3(c) and 3(d).

3.2. Solitary Wave Solutions

From Figure 1(c), we see that there is one homoclinic orbit of system (7) defined by connecting with saddle point and passing point when , , and , where . Its expression is

Substituting (16) into and integrating it along the homoclinic orbit yield equation

Completing (17) and using transformation (4), we can get a solitary wave solution of (3) as follows:

where is a new parametric variable and . The profiles of (18) are shown in Figures 4(a) and 4(b).

(a)  

(b)

(c)

(d)

(e)

(f)

(g)

(h)

Figure 4: Solitary waves of (3) for .

From Figure 1(h), we see that there is one homoclinic orbit of system (7) defined by connecting with saddle point and passing point when , , and , where . Its expression is

Substituting (19) into and integrating it along the homoclinic orbit yield equation

Completing (20) and using transformation (4), we can get a solitary wave solution of (3) as follows:

where . The profiles of (21) are shown in Figures 4(c) and 4(d).

From Figure 2(c), we see that there is one homoclinic orbit of system (7) defined by connecting with saddle point and passing point when , , and , where . Its expression is

Substituting (22) into and integrating it along the homoclinic orbit yield equation

Completing (23) and using transformation (4), we can get a solitary wave solution of (3) as follows:

where . The profiles of (24) are shown in Figures 4(e) and 4(f).

From Figure 2(j), we see that there is one homoclinic orbit of system (7) defined by connecting with saddle point and passing point when , , , where . Its expression is

Substituting (25) into and integrating it along the homoclinic orbit yield equation

Completing (26) and using transformation (4), we can get a solitary wave solution of (3) as follows:

where . The profiles of (27) are shown in Figures 4(g) and 4(h).

3.3. Smooth Periodic Wave Solutions

From Figure 1(d), we see that there is a periodic orbit of system (7) defined by enclosing the center point and passing points and when , , and . Its expression is

Substituting (28) into and integrating it along the periodic orbit yield equation

Completing (29) and using transformation (4), we can get a smooth periodic wave solution of (3) as follows:

where . The profiles of (30) are shown in Figures 5(a) and 5(b).

(a)  

(b)

(c)

(d)

Figure 5: Smooth periodic waves of (3) for .

From Figure 1(i), we see that there is a periodic orbit of system (7) defined by enclosing the center point and passing points and when , , and . Its expression is

Substituting (31) into and integrating it along the periodic orbit yield equation

Completing (32) and using transformation (4), we can get a smooth periodic wave solution of (3) which is the same as (30). The profiles are shown in Figures 5(c) and 5(d).

3.4. Compacton Solutions

For given , (or , ) in Figure 1(c), the level curve defined by is shown in Figure 6(a).

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Figure 6: The level curves defined by .

For given , in Figure 1(c), in Figure 1(d), and in Figures 1(e) and 1(j), respectively, the level curve defined by is shown in Figure 6(b).

For given , (or , ) in Figure 1(h),the level curve defined by is shown in Figure 6(c).

For given , in Figure 1(h), and in Figure 1(i), in Figures 1(e) and 1(j), respectively, the level curve defined by is shown in Figure 6(d).

For given in Figures 2(e), 2(f), 2(g), and 2(n), respectively, the level curve defined by is shown in Figure 6(e).

For given in Figures 2(g), 2(l), 2(m), and 2(n), respectively, the level curve defined by is shown in Figure 6(f).

From Figure 6(a), we see that there are a periodic orbit and an open curve of system (7) defined by when , , , , (or , and ). The open curve passes point and approaches the line ; its expression is

where , , and are three real roots of . For example, , , when , , , , and , and , , and when , , , , and .

Substituting (33) into and integrating it along the open curve yield equation

Completing (34) and using transformation (4), we can get the implicit representation of a compacton solution of (3) as follows:

where , , , and ; is Legendre’s incomplete elliptic integral of the third kind, and is the inverse function of the Jacobian elliptic function [15].

From Figure 6(b), we see that there are a periodic orbit and an open curve of system (7) defined by if and only if one of the following conditions holds:

(i), , , , and ,(ii), , , and ,(iii), , , and .

The open curve passes point and approaches the line ; its expression is

where , , and () are three real roots of . For example, , , and when , , and and , when , , , , and .

Substituting (36) into and integrating it along the open curve yield equation

Completing (37) and using transformation (4), we can get the implicit representation of a compacton solution of (3) as follows:

where , , , and .

From Figure 6(c), we see that there are a periodic orbit and an open curve of system (7) defined by when , , , , (or , and ). The open curve passes point and approaches the line ; its expression is

where , , and () are three real roots of . For example, , , and when , , , , and and , , and when , , , , and .

Substituting (39) into and integrating it along the open curve yield equation

Completing (40) and using transformation (4), we can get the implicit representation of a compacton solution of (3) as follow:

where , , , and .

From Figure 6(d), we see that there are a periodic orbit and an open curve of system (7) defined by if and only if one of the following conditions holds:

(i), , , , and ,(ii), , , and ,(iii), , , and .

The open curve passes point and approaches the line , its expression is

where , , and () are three real roots of . For example, , , and when , , , , and and , , when , , , , and .

Substituting (42) into and integrating it along the open curve yield equation

Completing (43) and using transformation (4), we can get the implicit representation of a compacton solution of (3) as follows:

where , , , and .

From Figure 6(e), we see that there are three open curves of system (7) defined by if and only if one of the following conditions holds:

(i), and ,(ii), and .

One of them passes point and approaches the line ; its expression is

where , , and are three real roots of . For example, , and when , , , and and , , , and when , , , , and .

Substituting (45) into and integrating it along the open curve yield equation

Completing (46) and using transformation (4), we can get the implicit representation of a compacton solution of (3) as follows:

where , , , and .

From Figure 6(f), we see that there are three open curves of system (7) defined by if and only if one of the following conditions holds:

(i), and ,(ii), and .

One of them passes point and approaches the line ; its expression is

where , , and are three real roots of . For example, , , and when , , , , and , and , , and when , , , , and .

Substituting (48) into and integrating it along the open curve yield equation

Completing (49) and using transformation (4), we can get the implicit representation of a compacton solution of (3) as follows:

where , , , and .

3.5. Periodic Cusp Wave Solutions

For given in Figure 1(e), the level curve defined by is shown in Figure 6(g).

For given in Figure 1(j), the level curve defined by is shown in Figure 6(h).

For given in Figure 2(e), the level curve defined by is shown in Figure 6(i).

For given in Figure 2(l), the level curve defined by is shown in Figure 6(j).

From Figures 6(g) and 6(h), we see that there are two heteroclinic orbits of system (7) defined by connecting with the saddle points and passing points , and , respectively, when , , and , where and . Their expressions are, respectively,

Substituting (51) into the and integrating it along the heteroclinic orbit, yields equation

Completing (53) and using transformation (4), we can get a periodic cusp wave solution of (3) as follows:

where , , , and .

Substituting (52) into and integrating it along the heteroclinic orbit yield equation

Completing (55) and using transformation (4), we can get a periodic cusp wave solution of (3) as follows:

where , , , and .

From Figure 6(i), we see that there are a heteroclinic orbit and an open curve of system (7) defined by when , and . The heteroclinic orbit connecting with the saddle points and passing point ; its expression is

where and .

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

Completing (58) and using transformation (4), we can get a periodic cusp wave solution of (3) as follows:

where , , and .

From Figure 6(j), we see that there are a heteroclinic orbit and an open curve of system (7) defined by when , and . The heteroclinic orbit connecting with the saddle points and passing point ; its expression is

where and .

Substituting (60) into and integrating it along the heteroclinic orbit yield equation

Completing (61) and using transformation (4), we can get a periodic cusp wave solution of (3) as follows:

where , , , and .

4. Numerical Simulations

In this section, we simulate the planar diagrams of the compactons and the periodic cusp waves of (3). From the derivations of (6) and (7), it is seen that if and are the parameter expressions of an orbit of system (7) and they satisfy and , then the diagram of and the integral curve of (6) with initial conditions and are the same. Therefore we can use the simulations of integral curves of (6) to test the validity of exact travelling wave solutions (35), (38), (41), (44), (47), (50), (54), (56), (59), and (62).

Example 1. Choose , , , , and satisfying the parametric conditions of Figure 6(a); that is , , , , and , and let in (33), then it follows that or . Taking , and as initial values of (6), using the Maple, we get the numerical simulation of the integral curve which corresponds to the open curve (33) as Figure 7(a).

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Figure 7: The numerical simulations of integral curves of (3).

Example 2. Choose , , , , and satisfying the parametric conditions of Figure 6(b); that is , , , , and , and let in (36); then it follows that or . Taking , and as initial values of (6), using the Maple, we get the numerical simulation of the integral curve which corresponds to the open curve (36) as Figure 7(b).

Example 3. Choose , , , , and satisfying the parametric conditions of Figure 6(c); that is , , , , and let in (39); then it follows that or . Taking and as initial values of (6), using the Maple, we get the numerical simulation of the integral curve which corresponds to the open curve (39) as Figure 7(c).

Example 4. Choose , , , , and satisfying the parametric conditions of Figure 6(d); that is , , , , and , and let in (42); then it follows that or . Taking and as initial values of (6), using the Maple, we get the numerical simulation of the integral curve which corresponds to the open curve (42) as Figure 7(d).

Example 5. Choose , , , , and satisfying the parametric conditions of Figure 6(e); that is , , , , , and let in (45); then it follows that or . Taking and as initial values of (6), using the Maple, we get the numerical simulation of the integral curve which corresponds to the open curve (45) as Figure 7(e).

Example 6. Choose , , , , and satisfying the parametric conditions of Figure 6(f); that is , , , , and , and let in (48); then it follows that or . Taking and as initial values of (6), using the Maple, we get the numerical simulation of the integral curve which corresponds to the open curve (48) as Figure 7(f).

Example 7. Choose , , , and satisfying the parametric conditions of Figure 6(g); that is , , , and . Let in (51); then it follows that or . Taking and as initial values of (6), using the Maple, we get the numerical simulation of the integral curve which corresponds to the heteroclinic orbit (51) as Figure 7(g). Let in (52), then it follows that or . Taking as initial values of (6), using the Maple, we get the numerical simulation of the integral curve which corresponds to the heteroclinic orbit (52) as Figure 7(h).

Example 8. Choose , , , and satisfying the parametric conditions of Figure 6(h); that is , , , and . Let in (51), then it follows that or . Taking and as initial values of (6), using the Maple, we get the numerical simulation of the integral curve which corresponds to the heteroclinic orbit (51) as Figure 7(i). Let in (52); then it follows that or . Taking as initial values of (6), using the Maple, we get the numerical simulation of the integral curve which corresponds to the heteroclinic orbit (52) as Figure 7(j).

Example 9. Choose , , , and satisfying the parametric conditions of Figure 6(i); that is , , , and and let in (57), then it follows that or . Taking and as initial values of (6), using the Maple, we get the numerical simulation of the integral curve which corresponds to the heteroclinic orbit (57) as Figure 7(k).

Example 10. Choose , , , and satisfying the parametric conditions of Figure 6(j); that is , , , and and let in (60); then it follows that or . Taking and as initial values of (6), using the Maple, we get the numerical simulation of the integral curve which corresponds to the heteroclinic orbit (60) as Figure 7(l).

Figure 7 implies that our theoretic results of the compacton and the periodic cusp wave solutions are identical with the numerical simulations.

5. Conclusion

In this paper, we studied the bifurcations of travelling wave solutions of (3) using the approach of dynamical system and obtained some exact peakon, compacton, solitary wave, smooth periodic wave, and periodic cusp wave solutions. Also, the compactons and the periodic cusp waves are simulated by approach of numerical simulation. The results of this paper have enriched results of references [9–11].

Acknowledgments

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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