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Journal of Applied Mathematics
Volume 2012, Article ID 363879, 25 pages
http://dx.doi.org/10.1155/2012/363879
Research Article

Periodic Wave Solutions and Their Limits for the Generalized KP-BBM Equation

1Department of Mathematics, South China University of Technology, Guangzhou 510640, China
2Department of Mathematics, Faculty of Sciences, Yuxi Normal University, Yuxi 653100, China

Received 17 April 2012; Accepted 15 May 2012

Academic Editor: Junjie Wei

Copyright © 2012 Ming Song and Zhengrong Liu. 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 use the bifurcation method of dynamical systems to study the periodic wave solutions and their limits for the generalized KP-BBM equation. A number of explicit periodic wave solutions are obtained. These solutions contain smooth periodic wave solutions and periodic blow-up solutions. Their limits contain periodic wave solutions, kink wave solutions, unbounded wave solutions, blow-up wave solutions, and solitary wave solutions.

1. Introduction

The Benjamin-Bona-Mahony (BBM) equation [1], has been proposed as a model for propagation of long waves where nonlinear dispersion is incorporated.

The Kadomtsov-Petviashvili (KP) equation [2] is given by which is a weekly two-dimensional generalization of the KdV equation in the sense that it accounts for slowly varying transverse perturbations of unidirectional KdV solitons moving along the -direction.

Wazwaz [3] presented the Kadomtsov-Petviashvili-Benjamin-Bona-Mahony (KP-BBM) equation and the generalized KP-BBM equation

Wazwaz [3, 4] obtained some solitons solution and periodic solutions of (1.3) by using the sine-cosine method and the extended tanh method. Abdou [5] used the extended mapping method with symbolic computation to obtain some periodic solutions of (1.3), solitary wave solution, and triangular wave solution. Song et al. [6] obtained exact solitary wave solutions of (1.3) by using bifurcation method of dynamical systems.

The aim of this paper is to study the traveling wave solutions and their phase portraits for (1.4) by using the bifurcation method and qualitative theory of dynamical systems [615]. Through some special phase orbits, we obtain a number of smooth periodic wave solutions and periodic blow-up solutions. Their limits contain periodic wave solutions, kink wave solutions, unbounded solutions, blow-up wave solutions, and solitary wave solutions.

The remainder of this paper is organized as follows. In Section 2, by using the bifurcation theory of planar dynamical systems, two phase portraits for the corresponding traveling wave system of (1.4) are given under different parameter conditions. In Section 3, we present our main results and their proofs. A short conclusion will be given in Section 4.

2. Phase Portraits

To derive our results, we give some preliminaries in this section. For given positive constant wave speed , substituting with into the generalized KP-BBM equation (1.4), it follows that

Integrating (2.1) twice and letting the first integral constant be zero, we have where is the second constant of integration.

Letting , we get the following planar system: where , and .

Obviously, the system (2.3) is a Hamiltonian system with Hamiltonian function where is Hamiltonian.

Now we consider the phase portraits of system (2.3). Set

Obviously, has three zero points, , and , which are given as follows:

It is easy to obtain two extreme points of as follows:

Letting then it is easily seen that is the extreme values of .

Let be one of the singular points of system (2.3), then the characteristic values of the linearized system of system (2.3) at the singular points are

From the qualitative theory of dynamical systems, we therefore know that(i)if , is a saddle point.(ii)if , is a center point.(iii)if , is a degenerate saddle point.

Therefore, we obtain the phase portraits of system (2.3) in Figures 1 and 2.

363879.fig.001
Figure 1: The phase portraits of system (2.3) when , .
363879.fig.002
Figure 2: The phase portraits of system (2.3) when , .

3. Main Results and Their Proofs

In this section, we state our main results.

Proposition 3.1. For given positive constants and , (1.4) has the following periodic wave solutions when and .
(1) If , we get eight periodic blow-up wave solutions two periodic wave solutions two kink wave solutions and two unbounded wave solutions
(2) If , we get four periodic blow-up wave solutions where a periodic wave solution a blow-up wave solution a solitary wave solution and an unbounded wave solution
(3) If , we get three blow-up wave solutions and a periodic blow-up wave solution where and are conjugate complex numbers.

Proof. (1) If , we will consider three kinds of orbits.
(i) From the phase portrait, we note that there are two special orbits and , which have the same Hamiltonian with that of the center point . In plane, the expressions of these two orbits are given as where and .
Substituting (3.18) into and integrating them along the two orbits and , it follows that
From (3.19) and noting that and , we get four periodic blow-up solutions and as (3.1) and (3.2).
(ii) From the phase portrait, we note that there are three special orbits , , and passing the points ,  ,  , and . In plane, the expressions of the orbit are given as where   and  .
Substituting (3.20) into and integrating them along , , and , we have
From (3.21) and noting that and , we get four periodic blow-up wave solutions , as (3.3), (3.4) and two periodic solutions as (3.5).
(iii) From the phase portrait, we see that there are two heteroclinic orbits and connected at saddle points and . In plane, the expressions of the heteroclinic orbits are given as
Substituting (3.22) into and integrating them along the heteroclinic orbits and , it follows that
From (3.23) and noting that and , we get two kink wave solutions as (3.6) and two unbounded solutions as (3.7).
(2) If , we set the largest solution of be , then we can get another two solutions of as follows:
(i) From the phase portrait, we see that there are two special orbits and , which have the same Hamiltonian with that of the center point . In -plane, the expressions of the orbits are given as where
Substituting (3.25) into and integrating them along the orbits, it follows that
From (3.27) and noting that and , we get two periodic blow-up wave solutions as (3.8).
(ii) From the phase portrait, we note that there are three special orbits , , and passing the points ,  , , and . In plane, the expressions of the orbit are given as where .
Substituting (3.28) into and integrating them along , , and , we have
From (3.29) and noting that and , we get two periodic blow-up wave solutions , as (3.9) and a periodic wave solution as (3.11).
(iii) From the phase portrait, we see that there are a spacial orbit , which passes the point , and a homoclinic orbit passing the saddle point . In plane, the expressions of the orbits are given as where
Substituting (3.30) into and integrating them along the orbits, it follows that
From (3.32) and noting that and , we get a blow-up solution as (3.12), a solitary wave solution as (3.13), and an unbounded wave solution as (3.14).
(3) If , we will consider two kinds of orbits.
(i) From the phase portrait, we see that there are two orbits and , which have the same Hamiltonian with the degenerate saddle point . In plane, the expressions of these two orbits are given as where
Substituting (3.33) into and integrating them along these two orbits and , it follows that
From (3.35) and noting that and , we get three blow-up solutions , , and as (3.15).
(ii) From the phase portrait, we see that there are two special orbits and passing the points and . In plane, the expressions of the orbits are given as where , and are conjugate complex numbers.
Substituting (3.36) into and integrating them along and , we have
From (3.37) and noting that and , we get a periodic blow-up wave solutions as (3.16).
Thus, we obtain the results given in Proposition 3.1.

Proposition 3.2. For given positive constants and , (1.4) has the following periodic wave solution when and .
(1) If , we get four periodic wave solutions where and two solitary wave solutions
(2) If , we get six periodic wave solutions where , , and are complex numbers.
And two solitary wave solutions
(3) If , we get two periodic wave solutions as follows: where , , and are complex numbers.
And a solitary wave solution

Proof. (1) If , we set
(i) From the phase portrait, we see that there are a closed orbit passing the points and . In plane, the expressions of the closed orbits are given as where , and .
Substituting (3.58) into and integrating them along the orbit , we have
From (3.59) and noting that and , we obtain the periodic wave solutions as (3.38) and as (3.39).
(ii) From the phase portrait, we see that there are two closed orbits and passing the points , , , and . In plane, the expressions of the closed orbits are given as where , and .
Substituting (3.60) into and integrating them along and , we have
From (3.61) and noting that and , we obtain the periodic wave solutions as (3.40) and as (3.41).
(iii) From the phase portrait, we see that there are two symmetric homoclinic orbits and connected at the saddle point . In plane, the expressions of the homoclinic orbits are given as where and .
Substituting (3.62) into and integrating them along the orbits and , we have
From (3.63) and noting that and , we obtain the solitary wave solutions as (3.43).
(2) If , we set the middle solution of be , then we can get another two solutions of as follows:
(i) From the phase portrait, we see that there are a closed orbit passing the points and . In plane, the expressions of the closed orbits are given as where , and are conjugate complex numbers.
Substituting (3.37) into and integrating them along , we have
From (3.66) and noting that and , we get a periodic wave solution as (3.44).
(ii) From the phase portrait, we note that there is a special orbit , which has the same Hamiltonian with that of . In plane, the expressions of the orbits are given as where
Substituting (3.67) into and integrating them along , it follows that
From (3.69) and noting that and , we get a periodic wave solution as (3.45).
(iii) From the phase portrait, we note that there are two closed orbits and passing the points , , , and . In plane, the expressions of the orbits are given as where .
Substituting (3.70) into and integrating them along and , we have
From (3.71) and noting that and , we get two periodic wave solutions as (3.46) and as (3.47).
(iv) From the phase portrait, we note that there is a special orbit passing the points and . In plane, the expressions of the orbit are given as where , and are conjugate complex numbers.
Substituting (3.72) into and integrating it along , we have
From (3.73) and noting that and , we get a periodic wave solution as (3.48).
If is a traveling wave solution, then is a traveling wave solution too. Taking and noting that , we get a periodic wave solution as (3.49).
(v) From the phase portrait, we note that there are two homoclinic orbits and connected at the saddle point . In plane, the expressions of the orbits are given as where
Substituting (3.74) into and integrating them along and , it follows that
From (3.76) and noting that and , we get two solitary wave solutions as (3.51) and as (3.52).
(3) If , we will consider two kinds of orbits.
(i) From the phase portrait, we note that there is a closed orbit passing the points and . In plane, the expressions of the orbit are given as where , and are conjugate complex numbers.
Substituting (3.77) into and integrating it along , we have
From (3.78) and noting that and , we get a periodic wave solutions as (3.53).
(ii) From the phase portrait, we note that there is a closed orbit passing the points and . In plane, the expressions of the orbit are given as where , and are conjugate complex numbers.
Substituting (3.79) into and integrating them along , we have
From (3.80) and noting that and , we get a periodic wave solutions as (3.54).
(iii) From the phase portrait, we see that there is a homoclinic orbit , which passes the degenerate saddle point . In plane, the expressions of the homoclinic orbit are given as where
Substituting (3.81) into and integrating them along , it follows that
From (3.83) and noting that and , we get a solitary wave solution as (3.56).
Thus, the derivation of Proposition 3.2 has been finished.

Proposition 3.3. For these solutions, the following are their relations.
(1) When tends to , the periodic blow-up wave solutions and tend to periodic blow-up wave solutions and , that is,
(2) When tends to , the periodic wave solutions tend to kink wave solutions , that is,
(3) When tends to , the periodic blow-up wave solutions tend to unbounded wave solutions , that is,
(4) When tends to , the periodic blow-up wave solution tends to periodic blow-up wave solution , that is,
(5) When tends to , the periodic blow-up wave solution tends to periodic blow-up wave solution , that is,
(6) When tends to , the periodic wave solution tends to solitary wave solution , that is,
(7) When tends to , the periodic blow-up wave solution tends to unbounded wave solution , that is,