# New Exact Solutions, Dynamical and Chaotic Behaviors for the Fourth-Order Nonlinear Generalized Boussinesq Water Wave Equation

Authors: Cheng Chen; Zuolei Wang
Journal: Advances in Mathematical Physics
 (2021)

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## Abstract

Based on the extended homogeneous balance method, the auto-Ba¨cklund transformation transformation is constructed and some new explicit and exact solutions are given for the fourth-order nonlinear generalized Boussinesq water wave equation. Then, the fourth-order nonlinear generalized Boussinesq water wave equation is transformed into the planer dynamical system under traveling wave transformation. We also investigate the dynamical behaviors and chaotic behaviors of the considered equation. Finally, the numerical simulations show that the change of the physical parameters will affect the dynamic behaviors of the system.


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## Body

## 1. Introduction

Many nonlinear phenomena can be described by nonlinear wave equations, where Boussinesq equation is one of the important mathematical models, which is widely used in shallow water and the percolation in horizontal porous media [1]. Boussinesq equation has many different forms [2–8]. Clarkson and Kruskal gave a new method for determining similarity reductions of Boussinesq equation in Ref. [2]. Nonclassical symmetry reduction can be used to reduce the Boussinesq equation in Ref. [3]. Based on the asymptotic expansion for small phase speed, an asymptotic semianalytical solution was found in Ref. [4]. Higher-order Boussinesq equations (fourth-order and sixth-order Boussinesq equations ) for two-way propagation of shallow water waves were studied in Ref. [5]. The generalized Boussinesq equation was reduced using Lie group and some exact solutions were obtained in Ref. [6]. The Hirota bilinear method is used to construct two soliton solutions for the (2+1)-dimensional Boussinesq equation, the (3+1)-dimensional Boussinesq equation, and the sixth-order Boussinesq equation. But the three variants of the Boussinesq equation are nonintegrable and do not admit N-soliton solutions in Ref. [7]. The generalized (2+1)-dimensional Boussinesq equation is investigated by the bifurcation method of dynamical systems, and some exact solutions were obtained in Ref. [8].In this work, we consider the fourth-order nonlinear generalized Boussinesq water wave equation:(1)utt−auxx−2bux2−2buuxx+cuxxxx=0,where u=ux,t is a real function and a, b, and c are real nonzero arbitrary constants [9–15].As far as the authors know, Lie symmetry method is used to analyze Equation (1) and some soliton wave solutions are obtained in Ref. [9]. Based on Hirota’s bilinear method, Qin got some rational solutions and interaction solutions of Equation (1) in Ref. [10]. The rogue wave and semirational solutions of Equation (1) are studied by Hirota’s bilinear method [11]. The general rogue wave solutions of arbitrary orders for Boussinesq equation were found by bilinear Kadomtsev-Petviashvili (KP) reduction method in Ref. [12]. Some travelling wave solutions of Equation (1) were given by extended tanh method and rational method [13, 14]; the solitary wave and shock wave solutions of Equation (1) were obtained by method of undetermined coefficients [15].Many powerful methods for solving exact solutions of partial differential equations have been developed such as Darboux transform [16, 17] and Lie group method [18–20]. The extended homogeneous balance method (EHBM) is an effective method solving nonlinear partial differential equations [21–23]. So, we apply EHBM to construct new solutions of Equation (1) in this paper.The analysis of bifurcation and chaos behavior is a very interesting nonlinear phenomenon, which has been applied in many fields, such as engineering, telecommunication, and ecology [24–27]. By analyzing the dynamic behavior of differential equation, we can study whether the periodic external perturbation will lead to the chaotic behavior of differential equation. So, we will study the dynamical behaviors and chaotic behaviors of Equation (1) in this paper.The rest of this paper is arranged as follows. In Section2, based on the extended homogeneous balance method, the auto-Ba¨cklund transformation is constructed and some new explicit and exact solutions of Equation (1) are obtained. In Section 3, based on the theory of plane dynamic system, dynamical behaviors and chaotic behaviors of Equation (1) are studied. Some conclusions are provided at the end of the paper.


## 2. Abundant New Explicit and Exact Solutions to Equation (1)

In terms of the idea of extended homogeneous balance method [21, 22], we assume that the solution of Equation (1) has the following form:
(2)ux,t=∂m+nfϕ∂xm∂tn+u0=fm+nϕxmϕtn+⋯,where u0=u0x,t is the arbitrary known seed solution and m, n, and function fϕ are to be determined later.From (2), we obtain
(3)uxxxx=fm+n+4ϕxm+4ϕtn+⋯,ux2=fm+n+1fm+n+1ϕx2m+2ϕt2n+⋯,uuxx=fm+nfm+n+2ϕx2m+2ϕt2n+⋯.According to the homogeneous principle method [21, 22], balancing the highest-order derivative term uxxxx and the highest-order nonlinear term, ux2,uuxx can be obtained
(4)2m+2=2m+2=m+4,n=2n=2n,which gives
(5)m=2,n=0.Thus, (2) can be rewritten as follows
(6)ux,t=∂2fϕ∂x2+u0=f′′ϕx2+f′ϕxx+u0.From (6), it is easy to deduce that
(7)utt=f4ϕt2ϕx2+f3ϕttϕx2+4f3ϕtϕxϕxt+2f′′ϕxt2+2f′′ϕxϕxtt+f3ϕt2ϕxx+f′′ϕxxϕtt+f′′ϕtϕxxt+f′ϕxxtt+u0tt,(8)uxx=f4ϕx4+6f3ϕx2ϕxx+3f′′ϕxx2+4f′′ϕxϕxxx+f′ϕxxxx+u0xx,(9)ux2=f32ϕx6+9f′′2ϕx2ϕxx2+f′2ϕxxx2+u0x2+6f3f′′ϕx4ϕxx+2f3f′ϕx3ϕxxx+2u0xf3ϕx3+6f′′f′ϕxϕxxϕxxx+6u0xf′′ϕxϕxx+2u0xf′ϕxxx,(10)uuxx=f′′f4ϕx6+6f′′f3ϕx4ϕxx+3f′′2ϕx2ϕxx2+4f′′2ϕx3ϕxxx+f′f′′ϕx2ϕxxxx+f′′ϕx2u0xx+f′f4ϕx4ϕxx+6f′f3ϕx2ϕxx2+3f′f′′ϕxx3+4f′f′′ϕxϕxxϕxxx+f′2ϕxxϕxxxx+f′ϕxxu0xx+f4u0ϕx4+6f3u0ϕx2ϕxx+3f′′u0ϕxx2+4f′′u0ϕxϕxxx+f′u0ϕxxxx+u0u0xx,(11)uxxxx=f6ϕx6+15f5ϕx4ϕxx+45f4ϕx2ϕxx2+20f4ϕx3ϕxxx+15f3ϕxx3+60f3ϕxϕxxϕxxx+15f3ϕx2ϕxxxx+10f′′ϕxx2+15f′′ϕxxϕxxxx+6f′′ϕxϕxxxx+f′ϕxxxxxx+u0xxxx.Now, substituting (7)–(11) into Equation (1) and simplifying, it yields
(12)ϕx6−2bf32−2bf4f′′+cf6+ϕx4−af4−24bf′′f3+2bf′f4−15cf5ϕxx−2bu0f4+ϕx3−4bf′f3−8bf′2+20cf4ϕxxx−4bf3u0x+ϕx2f4ϕt2+f3ϕtt−6a+12bu0f3ϕxx+−24bf′′2+2bf′f3+45cf4ϕxx2+−2bf′f′′+15cf3ϕxxxx−2bf′′u0xx+ϕx4f3ϕtϕxt+2f′′ϕxtt+−4af′′−20bf′f′′ϕxx+60cf3ϕxx−8bu0f′′ϕxxx+6cf′′ϕxxxx−12bu0xf′′ϕxx,(13)ϕxx3−6bf′f′′+15cf3−3a+6bu0f′′ϕxx2+ϕxxf3ϕt2−2bf′2ϕxxxx+15cf′′ϕxxxx+f′′ϕtt−2bf′u0xx+ϕxxx2−2bf′2+10cf′′−4bu0xf′ϕxxx−a+2bu0f′ϕxxxx+2f′′ϕxt2+f′ϕttxx+2f′′ϕtϕxxt+cf′ϕxxxxxx+u0tt−au0xx+cu0xxxx−2bu0x2−2bu0u0xx=0.Setting the coefficient of the termϕx6 in Equation (13) to zero, we obtain a nonlinear ordinary differential equation for function fϕ:
(14)−2bf32−2bf4f′′+cf6=0.Integrating Equation (14) once and neglecting the constant of integration gives
(15)−2bf3f′′+cf5=0.Again integrating Equation (15) and letting the constant of integration be zero, we obtain
(16)−bf′′2+cf4=0,which has particular solution
(17)fϕ=rlnϕ,where r=−6c/b.According to (17), we get the following results:
(18)f′2=−rf′′,f′′f′=−r2f3,f3f′=−r3f4,(19)f4f′=−r4f5,f′′2=−r6f4,f′′f3=−r12f5.By (6) and (17), we obtain the auto-Ba¨cklund transformation of Equation (1) as follows:
(20)ux,t=−rϕx2ϕ2+rϕxxϕ+u0.Letting the seed solution(21)u0x,t=0,and using results (19), then Equation (13) can be simplified as
(22)f4−aϕx2+4cϕxϕxxx+ϕt2−3cϕxx2ϕx2+f3ϕtt−6aϕxx+9cϕxxxxϕx2+4ϕxϕtϕxt−3cϕxx3+ϕxxϕt2+f′′ϕxt2+2ϕxϕxtt+ϕttϕxx+2ϕtϕxxt−3aϕxx2−4aϕxϕxxx+6cϕxϕxxxxx−2cϕxxx2+3cϕxxϕxxxx+f′ϕxxtt−aϕxxxx+cϕxxxxxx=0.Setting the coefficients off4, f3, f′′, and f′ in Equation (22) to zero yields a set of partial differential equations for ϕx,t as follows:
(23)ϕx2−aϕx2+4cϕxϕxxx+ϕt2−3cϕxx2=0,(24)ϕtt−6aϕxx+9cϕxxxxϕx2+4ϕxϕtϕxt−3cϕxx3+ϕxxϕt2=0,(25)ϕxt2+2ϕxϕxtt+ϕttϕxx+2ϕtϕxxt−3aϕxx2−4aϕxϕxxx+6cϕxϕxxxxx−2cϕxxx2+3cϕxxϕxxxx=0,(26)ϕtt−aϕxx+cϕxxxxxx=0.Here, Equations (24) and (25) can be rewritten as
(27)ϕx2ϕtt−aϕxx+cϕxxxx+ϕxx−aϕx2+4cϕxϕxxx+ϕt2−3cϕxx2+2ϕx−aϕx2+4cϕxϕxxx+ϕt2−3cϕxx2x=0,(28)ϕttϕtt−aϕxx+cϕxxxx+2ϕxϕtt−aϕxx+cϕxxxxx+−aϕx2+4cϕxϕxxx+ϕt2−3cϕxx2xx=0.By analysis of Equations (27) and (28), we find that Equations (23)–(26) are satisfied automatical under the following conditions:
(29)ϕtt−aϕxx+cϕxxxx=0,(30)−aϕx2+4cϕxϕxxx+ϕt2−3cϕxx2=0.To obtain some exact solutions of Equations (29) and (30), we assume the solutions of the form:
(31)ϕx,t=A+Bsinξexpη,where ξ=Kx+Ωt+ξ0,η=kx+ωt+η0;A, B, K, Ω, ξ0, k, ω, and η0 are constants which are to be determined.Substituting (31) into Equations (29) and (30) and simplifying, it leads to a system of nonlinear algebraic equations with respect to K, Ω, k,and ω as follows
(32)−Ω2+ω2+aK2−ak2+cK4−6cK2k2+ck4=0,2Ωω−2aKk−4ckK3+4cKk3=0,−2aKk+2Ωω+12c−4kK3−12c−4Kk3=0,−ak2+ω2+ck4−6cK2k2+aK2−Ω2+cK4=0,Ω2−aK2−4cK4=0.By solving the above equations, the following sets of solutions are obtainedCase 1.
K=0,Ω=0,k=k,ω=ω, where k≠0,c≠1/3,ω2+ck4−ak2=0,ac>0.Case 2.
k=0,Ω=0,K=−a/4c,ω=±3a2/16c, where c>0,a<0,c≠1/3.Case 3.
k=0,Ω=0,K=−−a/4c,ω=±3a2/16c, where c>0,a<0,c≠1/3.Case 4.
Ω=Ω,k=−Ω2/4c4,K=−−Ω2/4c4,ω=2Ω, where c<0,c≠1/3.Case 5.
Ω=Ω,k=−−Ω2/4c4,K=−Ω2/4c4,ω=2Ω, where c<0,c≠1/3.Case 6.
Ω=Ω,k=3Ω2/4c4,K=3Ω2/4c4,ω=−2Ω, where c>0,c≠1/3.Case 7.
Ω=Ω,k=−3Ω2/4c4,K=−3Ω2/4c4,ω=−2Ω, where c>0,c≠1/3.According to (20), (21), and (31), the solution of the fourth-order nonlinear generalized Boussinesq water wave equation is as follows:
(33)ux,t=−rϕx2ϕ2+rϕxxϕ=6cB2exp2ηKcosξ+ksinξ2bA+Bsinξexpη2−6cBexpηk2−K2sinξ+2KkcosξbA+Bsinξexpη,where ξ=Kx+Ωt+ξ0,η=kx+ωt+η0.For Case1, the exact explicit solutions to the fourth-order nonlinear generalized Boussinesq water wave equation are obtained as follows:
(34)ux,t=6cB2k2sin2ξ0exp2kx+ωt+η0bA+Bsinξ0exp2kx+ωt+η02−6cBk2sinξ0expkx+ωt+η0bA+Bsinξ0exp2kx+ωt+η0.IfA=1,Bsinξ0=1, Equation (34) is reduced to the solitary wave solutions as follows:
(35)ux,t=−3ck22bsech212kx+ωt+η0.For Case2, the exact explicit solutions to the fourth-order nonlinear generalized Boussinesq water wave equation are obtained as follows:
(36)ux,t=−3aB2exp±3a2/16ct+η0exp±3a2/16ct+η0+Asin−a/4cx+ξ02cbA+Bsin−a/4cx+ξ0exp±3a2/16ct+η02.IfA=0, Equation (36) is reduced to
(37)ux,t=−3a2bcsc2−a4cx+ξ0,which are periodic wave solutions.For Case3, the exact explicit solutions to the fourth-order nonlinear generalized Boussinesq water wave equation are obtained as follows:
(38)ux,t=−3aB2exp±3a2/16ct+η0exp±3a2/16ct+η0+Asin−−a/4cx+ξ02cbA+Bsin−−a/4cx+ξ0exp±3a2/16ct+η02.IfA=0, Equation (38) is reduced to Equation (37).For Case4, the exact explicit solutions to the fourth-order nonlinear generalized Boussinesq water wave equation are obtained as follows:
(39)ux,t=−6cB−Ω2/4c4exp−Ω2/4c4x+2Ωt+η0cos−−Ω2/4c4x+Ωt+ξ0−sin−−Ω2/4c4x+Ωt+ξ0bA+Bsin−−Ω2/4c4x+Ωt+ξ0exp−Ω2/4c4x+2Ωt+η02+12cB−Ω2/4cexp−Ω2/4c4x+2Ωt+η0cos−−Ω2/4c4x+Ωt+ξ0bA+Bsin−−Ω2/4c4x+Ωt+ξ0exp−Ω2/4c4x+2Ωt+η0.For Case5, the exact explicit solutions to the fourth-order nonlinear generalized Boussinesq water wave equation are obtained as follows:
(40)ux,t=6cB−Ω2/4c4exp−−Ω2/4c4x+2Ωt+η0cos−Ω2/4c4x+Ωt+ξ0−sin−Ω2/4c4x+Ωt+ξ0bA+Bsin−Ω2/4c4x+Ωt+ξ0exp−−Ω2/4c4x+2Ωt+η02+12cB−Ω2/4cexp−−Ω2/4c4x+2Ωt+η0cos−Ω2/4c4x+Ωt+ξ0bA+Bsin−Ω2/4c4x+Ωt+ξ0exp−−Ω2/4c4x+2Ωt+η0.For Case6, the exact explicit solutions to the fourth-order nonlinear generalized Boussinesq water wave equation are obtained as follows:
(41)ux,t=6cB3Ω2/4c4exp3Ω2/4c4x−2Ωt+η0cos3Ω2/4c4x+Ωt+ξ0−sin3Ω2/4c4x+Ωt+ξ0bA+Bsin−Ω2/4c4x+Ωt+ξ0exp3Ω2/4c4x−2Ωt+η02−12cB3Ω2/4cexp3Ω2/4c4x−2Ωt+η0cos3Ω2/4c4x+Ωt+ξ0bA+Bsin3Ω2/4c4x+Ωt+ξ0exp3Ω2/4c4x−2Ωt+η0.For Case7, the exact explicit solutions to the fourth-order nonlinear generalized Boussinesq water wave equation are obtained as follows:
(42)ux,t=−6cB3Ω2/4c4exp−3Ω2/4c4x−2Ωt+η0cos−3Ω2/4c4x+Ωt+ξ0−sin−3Ω2/4c4x+Ωt+ξ0bA+Bsin−−Ω2/4c4x+Ωt+ξ0exp−3Ω2/4c4x−2Ωt+η02−12cB3Ω2/4cexp−3Ω2/4c4x−2Ωt+η0cos−3Ω2/4c4x+Ωt+ξ0bA+Bsin−3Ω2/4c4x+Ωt+ξ0exp−3Ω2/4c4x−2Ωt+η0.Remark 1.
Equations (29) and (30) may also have other forms of solutions as follows:
(43)ϕx,t=A+Bcosξexpη,ϕx,t=A+Bsinhξexpη,ϕx,t=A+Bcoshξexpη,where ξ=Kx+Ωt+ξ0,η=kx+ωt+η0;A,B,K,Ω,ξ0,k,ω,andη0 are constants which are to be determined.


## 3. Dynamical Behaviors and Chaotic Behaviors of Equation (1)

In this section, based on the theory of the plane dynamic system, the bifurcation analysis of the unperturbed dynamic system which is obtained by traveling wave transformation (special combinations of Lie point symmetries) is carried out. The periodic perturbed term is added to the obtained unperturbed dynamic system, and the chaotic behaviors of perturbed system is analyzed under different values of the physical parameters.


### 3.1. Bifurcation and Phase Portraits of Equation (1)

In Ref. [9], Lie point symmetries admitted by Equation (1) were given as follows:
(44)V1=∂∂t,V2=∂∂x,V3=2bt∂∂t+bx∂∂x−a+2bu∂∂u,which form a three-dimensional Lie algebra.We consider the Lie point symmetry as follows:(45)k1V1+k2V2=k1∂∂t+k2∂∂x,where k1,k2 are constants.The corresponding characteristic equation is(46)dtk1=dxk2=du0.Solving the above characteristic equation, we obtain the group invariant solution:(47)ux,t=Xξ,where ξ=k1x−k2t is the similarity invariant. This is a traveling wave transformation, which we will use to analyze dynamic behaviors of the equation.Under the traveling wave transformation, Equation (1) is reduced to
(48)k22−ak12X′′−2bk12X′2+XX′′+ck14X4=0,where the ``′” denotes the differentiation with respect to the variable ξ.Integrating Equation (48) twice with respect to ξ and vanishing the constants of integration yields
(49)X′′=αX2+βX.where α=b/ck12,β=ak12−k22/ck14.LettingX′=Y, Equation (49) is equivalent to the two-dimensional plane autonomous system:
(50)dXdξ=Y,dYdξ=αX2+βX.Obviously, system (50) is an integrable Hamiltonian system. Applying the first integral, we get Hamiltonian functions as follows:
(51)12Y2−α3X3−β2X2=h,where h is the constant of integration.By solving the following system:(52)Y=0,αX2+βX=0,equilibrium points of system (52) can be obtained as P10,0 and P2−β/α,0.LetMPi denote the coefficient matrix of the linearized system of system (50) at equilibrium points PiXi,0i=1,2 as follows:
(53)MPi=012αXi+β0,where the trace of MPi is zero and the determinant of M is JPi=detMPi=−2αXi+β. Then, we obtain detMP1=−β and detMP2=β.By the theory of planar dynamical systems [24–30], we draw the following conclusion as in Table 1. The different phase portraits of system (50) are shown in Figures 1 and 2.Table 1
Information of equilibrium points.
No.Values of parametersdetMP1detMP2P1P2Phase portraits1c>0,a<0JP1>0JP2<0center pointsaddle pointFigure1(a)c>0,a>0,∣k2∣>a∣k1∣JP1>0JP2<0center pointsaddle pointFigure1(b)c<0,a>0,∣k2∣<a∣k1∣JP1>0JP2<0center pointsaddle pointFigure1(c)c<0,a<0JP1<0JP2>0saddle pointcenter pointFigure2(a)c>0,a>0,∣k2∣<a∣k1∣JP1<0JP2>0saddle pointcenter pointFigure2(b)c<0,a>0,∣k2∣>a∣k1∣JP1<0JP2>0saddle pointcenter pointFigure2(c)Figure 1
Phase portraits of system (50).
(a)(b)(c)Figure 2
Phase portraits of system (50).
(a)(b)(c)


### 3.2. Chaotic Behaviors of Equation (1)

In this section, the dynamics of system (50) perturbed by periodic term is to be investigated via numerical simulations. For this purpose, suppose that periodic term is g0cosωξ. Letting Z=ωξ, system (50) is modified as following three-dimensional system:
(54)dXdξ=Y,dYdξ=αX2+BβX+g0cosZ,dZdξ=ω.In the simulations, choosing parametersα=2,β=−8, the effects of amplitude and frequency on the dynamics of system (54) are to be discussed.Firstly, the effect of amplitude is considered. For this end, frequencyω is fixed while g0 is chosen as different values. The corresponding phase portraits are depicted in Figures 3 and 4. By analyzing Figures 3 and 4, we can get the following results.Case 1.
Wheng0=0. System (54) is undisturbed by periodic signal, and it is provided with periodic solution (see Figures 3(a) and 4(a)).Case 2.
Wheng0≠0. Figure 3 means that the system (54) is disturbed by periodic signal, and system (54) presents limit cycles for low frequency (see Figures 3(b)–3(d)). Figure 4 shows that, when system (54) is disturbed by periodic signal with higher frequency, it can appear as limit cycles (see Figures 4(b) and 4(c)) and chaotic phenomenon (see Figure 4(d))with the change of amplitude.Figure 3
Phase portraits of system (54) with ω=0.01 and different values of g0. (a) g0=0, (b) g0=0.01, (c) g0=0.1, and (d) g0=5.
(a)(b)(c)(d)Figure 4
Phase portraits of system (54) with ω=10 and different values of g0. (a) g0=0, (b) g0=0.01, (c) g0=0.1, and (d) g0=10.
(a)(b)(c)(d)Consequently, Figures3 and 4 indicate that the amplitude has much effect on the dynamics of system (54).Secondly,g0 taken as a constant, and ω is chosen as different values. The corresponding phase portraits are calculated and shown in Figures 5 and 6. In Figure 5, it is obvious to see that, for small amplitude g0=0.1, with the change of frequency, system (54) has various phenomenon, such as chaos (see Figures 5(a) and 5(b)) and limit cycle (Figures 5(c) and 5(d)). Figure 6 suggests that, for small amplitude g0=5, system (54) mainly shows chaos.Figure 5
Phase portraits of system (54) with g0=0.1 and different values of ω. (a) ω=0.1. (b) ω=1. (c) ω=10. (d) ω=100.
(a)(b)(c)(d)Figure 6
Phase portraits of system (54) with g0=5 and different values of ω. (a) ω=0.001. (b) ω=0.1. (c) ω=1. (d) ω=10.
(a)(b)(c)(d)


## 3.1. Bifurcation and Phase Portraits of Equation (1)

In Ref. [9], Lie point symmetries admitted by Equation (1) were given as follows:
(44)V1=∂∂t,V2=∂∂x,V3=2bt∂∂t+bx∂∂x−a+2bu∂∂u,which form a three-dimensional Lie algebra.We consider the Lie point symmetry as follows:(45)k1V1+k2V2=k1∂∂t+k2∂∂x,where k1,k2 are constants.The corresponding characteristic equation is(46)dtk1=dxk2=du0.Solving the above characteristic equation, we obtain the group invariant solution:(47)ux,t=Xξ,where ξ=k1x−k2t is the similarity invariant. This is a traveling wave transformation, which we will use to analyze dynamic behaviors of the equation.Under the traveling wave transformation, Equation (1) is reduced to
(48)k22−ak12X′′−2bk12X′2+XX′′+ck14X4=0,where the ``′” denotes the differentiation with respect to the variable ξ.Integrating Equation (48) twice with respect to ξ and vanishing the constants of integration yields
(49)X′′=αX2+βX.where α=b/ck12,β=ak12−k22/ck14.LettingX′=Y, Equation (49) is equivalent to the two-dimensional plane autonomous system:
(50)dXdξ=Y,dYdξ=αX2+βX.Obviously, system (50) is an integrable Hamiltonian system. Applying the first integral, we get Hamiltonian functions as follows:
(51)12Y2−α3X3−β2X2=h,where h is the constant of integration.By solving the following system:(52)Y=0,αX2+βX=0,equilibrium points of system (52) can be obtained as P10,0 and P2−β/α,0.LetMPi denote the coefficient matrix of the linearized system of system (50) at equilibrium points PiXi,0i=1,2 as follows:
(53)MPi=012αXi+β0,where the trace of MPi is zero and the determinant of M is JPi=detMPi=−2αXi+β. Then, we obtain detMP1=−β and detMP2=β.By the theory of planar dynamical systems [24–30], we draw the following conclusion as in Table 1. The different phase portraits of system (50) are shown in Figures 1 and 2.Table 1
Information of equilibrium points.
No.Values of parametersdetMP1detMP2P1P2Phase portraits1c>0,a<0JP1>0JP2<0center pointsaddle pointFigure1(a)c>0,a>0,∣k2∣>a∣k1∣JP1>0JP2<0center pointsaddle pointFigure1(b)c<0,a>0,∣k2∣<a∣k1∣JP1>0JP2<0center pointsaddle pointFigure1(c)c<0,a<0JP1<0JP2>0saddle pointcenter pointFigure2(a)c>0,a>0,∣k2∣<a∣k1∣JP1<0JP2>0saddle pointcenter pointFigure2(b)c<0,a>0,∣k2∣>a∣k1∣JP1<0JP2>0saddle pointcenter pointFigure2(c)Figure 1
Phase portraits of system (50).
(a)(b)(c)Figure 2
Phase portraits of system (50).
(a)(b)(c)


## 3.2. Chaotic Behaviors of Equation (1)

In this section, the dynamics of system (50) perturbed by periodic term is to be investigated via numerical simulations. For this purpose, suppose that periodic term is g0cosωξ. Letting Z=ωξ, system (50) is modified as following three-dimensional system:
(54)dXdξ=Y,dYdξ=αX2+BβX+g0cosZ,dZdξ=ω.In the simulations, choosing parametersα=2,β=−8, the effects of amplitude and frequency on the dynamics of system (54) are to be discussed.Firstly, the effect of amplitude is considered. For this end, frequencyω is fixed while g0 is chosen as different values. The corresponding phase portraits are depicted in Figures 3 and 4. By analyzing Figures 3 and 4, we can get the following results.Case 1.
Wheng0=0. System (54) is undisturbed by periodic signal, and it is provided with periodic solution (see Figures 3(a) and 4(a)).Case 2.
Wheng0≠0. Figure 3 means that the system (54) is disturbed by periodic signal, and system (54) presents limit cycles for low frequency (see Figures 3(b)–3(d)). Figure 4 shows that, when system (54) is disturbed by periodic signal with higher frequency, it can appear as limit cycles (see Figures 4(b) and 4(c)) and chaotic phenomenon (see Figure 4(d))with the change of amplitude.Figure 3
Phase portraits of system (54) with ω=0.01 and different values of g0. (a) g0=0, (b) g0=0.01, (c) g0=0.1, and (d) g0=5.
(a)(b)(c)(d)Figure 4
Phase portraits of system (54) with ω=10 and different values of g0. (a) g0=0, (b) g0=0.01, (c) g0=0.1, and (d) g0=10.
(a)(b)(c)(d)Consequently, Figures3 and 4 indicate that the amplitude has much effect on the dynamics of system (54).Secondly,g0 taken as a constant, and ω is chosen as different values. The corresponding phase portraits are calculated and shown in Figures 5 and 6. In Figure 5, it is obvious to see that, for small amplitude g0=0.1, with the change of frequency, system (54) has various phenomenon, such as chaos (see Figures 5(a) and 5(b)) and limit cycle (Figures 5(c) and 5(d)). Figure 6 suggests that, for small amplitude g0=5, system (54) mainly shows chaos.Figure 5
Phase portraits of system (54) with g0=0.1 and different values of ω. (a) ω=0.1. (b) ω=1. (c) ω=10. (d) ω=100.
(a)(b)(c)(d)Figure 6
Phase portraits of system (54) with g0=5 and different values of ω. (a) ω=0.001. (b) ω=0.1. (c) ω=1. (d) ω=10.
(a)(b)(c)(d)


## 4. Conclusions

The extended homogeneous balance method is an effective method solving nonlinear partial differential equations. We applied it to obtain the auto-Ba¨cklund transformation transformation and some new exact explicit solutions for the fourth-order nonlinear generalized Boussinesq water wave equation. Using the theory of plane dynamic system, the dynamical behavior analysis of the perturbed dynamic system and the chaotic behaviors of the perturbed system are analyzed. The change of the physical parameters will affect the dynamic behavior of the dynamic system.



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