new exact solutions, dynamical and chaotic behaviors for

Research ArticleNew Exact Solutions, Dynamical and Chaotic Behaviors for theFourth-Order Nonlinear Generalized Boussinesq WaterWave Equation

Cheng Chen 1 and Zuolei Wang 2

1School of Science, Xi’an University of Posts and Telecommunications, Xi’an, Shaanxi 710121, China2School of Mathematics and Statistics, Yancheng Teachers University, Jiangsu 224002, China

Correspondence should be addressed to Cheng Chen; [email protected] and Zuolei Wang; [email protected]

Received 19 May 2021; Revised 6 August 2021; Accepted 25 August 2021; Published 24 September 2021

Academic Editor: Stephen C. Anco

Copyright © 2021 Cheng Chen and Zuolei Wang. This is an open access article distributed under the Creative CommonsAttribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original workis properly cited.

Based on the extended homogeneous balance method, the auto-B€acklund transformation transformation is constructed and somenew explicit and exact solutions are given for the fourth-order nonlinear generalized Boussinesq water wave equation. Then, thefourth-order nonlinear generalized Boussinesq water wave equation is transformed into the planer dynamical system undertraveling 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.

1. Introduction

Many nonlinear phenomena can be described by nonlinearwave equations, where Boussinesq equation is one of theimportant mathematical models, which is widely used inshallow water and the percolation in horizontal porousmedia [1]. Boussinesq equation has many different forms[2–8]. Clarkson and Kruskal gave a new method for deter-mining similarity reductions of Boussinesq equation in Ref.[2]. Nonclassical symmetry reduction can be used to reducethe Boussinesq equation in Ref. [3]. Based on the asymptoticexpansion for small phase speed, an asymptotic semianalyti-cal solution was found in Ref. [4]. Higher-order Boussinesqequations (fourth-order and sixth-order Boussinesq equa-tions ) for two-way propagation of shallow water waves werestudied in Ref. [5]. The generalized Boussinesq equation wasreduced using Lie group and some exact solutions wereobtained in Ref. [6]. The Hirota bilinear method is used toconstruct two soliton solutions for the (2 + 1)-dimensional

Boussinesq equation, the (3 + 1)-dimensional Boussinesqequation, and the sixth-order Boussinesq equation. But thethree variants of the Boussinesq equation are nonintegrableand do not admit N-soliton solutions in Ref. [7]. The gener-alized (2 + 1)-dimensional Boussinesq equation is investi-gated by the bifurcation method of dynamical systems, andsome exact solutions were obtained in Ref. [8].

In this work, we consider the fourth-order nonlineargeneralized Boussinesq water wave equation:

utt − auxx − 2bu2x − 2buuxx + cuxxxx = 0, ð1Þ

where u = uðx, tÞ is a real function and a, b, and c are realnonzero arbitrary constants [9–15].

As far as the authors know, Lie symmetry method is usedto analyze Equation (1) and some soliton wave solutions areobtained in Ref. [9]. Based on Hirota’s bilinear method, Qingot some rational solutions and interaction solutions ofEquation (1) in Ref. [10]. The rogue wave and semirational

HindawiAdvances in Mathematical PhysicsVolume 2021, Article ID 8409615, 13 pageshttps://doi.org/10.1155/2021/8409615

https://orcid.org/0000-0001-6134-4956

https://orcid.org/0000-0002-3442-2717

https://creativecommons.org/licenses/by/4.0/




https://doi.org/10.1155/2021/8409615

solutions of Equation (1) are studied by Hirota’s bilinearmethod [11]. The general rogue wave solutions of arbitraryorders for Boussinesq equation were found by bilinearKadomtsev-Petviashvili (KP) reduction method in Ref.[12]. Some travelling wave solutions of Equation (1) weregiven 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 coeffi-cients [15].

Many powerful methods for solving exact solutions ofpartial differential equations have been developed such asDarboux transform [16, 17] and Lie group method[18–20]. The extended homogeneous balance method(EHBM) is an effective method solving nonlinear partial dif-ferential equations [21–23]. So, we apply EHBM to constructnew solutions of Equation (1) in this paper.

The analysis of bifurcation and chaos behavior is avery interesting nonlinear phenomenon, which has beenapplied in many fields, such as engineering, telecommuni-cation, and ecology [24–27]. By analyzing the dynamicbehavior of differential equation, we can study whetherthe periodic external perturbation will lead to the chaoticbehavior of differential equation. So, we will study thedynamical behaviors and chaotic behaviors of Equation(1) in this paper.

The rest of this paper is arranged as follows. In Section 2,based on the extended homogeneous balance method, theauto-B€acklund transformation is constructed and somenew explicit and exact solutions of Equation (1) areobtained. In Section 3, based on the theory of plane dynamicsystem, dynamical behaviors and chaotic behaviors of Equa-tion (1) are studied. Some conclusions are provided at theend of the paper.

2. Abundant New Explicit and ExactSolutions to Equation (1)

In terms of the idea of extended homogeneous balancemethod [21, 22], we assume that the solution of Equation(1) has the following form:

u x, tð Þ = ∂ m+nð Þ f ϕð Þ∂xm∂tn

+ u0 = f m+nð Þϕmx ϕnt +⋯, ð2Þ

where u0 = u0ðx, tÞ is the arbitrary known seed solution andm, n, and function f ðϕÞ are to be determined later.

From (2), we obtain

uxxxx = f m+n+4ð Þϕm+4x ϕnt +⋯,

u2x = f m+n+1ð Þ f m+n+1ð Þϕ2m+2x ϕ2nt +⋯,

uuxx = f m+nð Þ f m+n+2ð Þϕ2m+2x ϕ2nt +⋯:

ð3Þ

According to the homogeneous principle method [21,22], balancing the highest-order derivative term uxxxx and

the highest-order nonlinear term, u2x , uuxx can beobtained

2m + 2 = 2m + 2 =m + 4, n = 2n = 2n, ð4Þ

which gives

m = 2, n = 0: ð5Þ

Thus, (2) can be rewritten as follows

u x, tð Þ = ∂2 f ϕð Þ∂x2

+ u0 = f ′′ϕ2x + f ′ϕxx + u0: ð6Þ

From (6), it is easy to deduce that

utt = f 4ð Þϕ2t ϕ2x + f 3ð Þϕttϕ

2x + 4f 3ð Þϕtϕxϕxt

+ 2f ′′ϕ2xt + 2f ′′ϕxϕxtt + f 3ð Þϕ2t ϕxx+ f ′′ϕxxϕtt + f ′′ϕtϕxxt + f ′ϕxxtt + u0tt ,

ð7Þ

uxx = f 4ð Þϕ4x + 6f 3ð Þϕ2xϕxx + 3f ′′ϕ2xx+ 4f ′′ϕxϕxxx + f ′ϕxxxx + u0xx,

ð8Þ

u2x = f 3ð Þ� �2

ϕ6x + 9 f ′′� �2

ϕ2xϕ2xx + f ′

� �2ϕ2xxx

+ u20x + 6f 3ð Þ f ′′ϕ4xϕxx + 2f 3ð Þ f ′ϕ3xϕxxx+ 2u0x f 3ð Þϕ3x + 6f ′′f ′ϕxϕxxϕxxx+ 6u0x f ′′ϕxϕxx + 2u0x f ′ϕxxx,

ð9Þ

uuxx = f ′′f 4ð Þϕ6x + 6f ′′f 3ð Þϕ4xϕxx + 3 f ′′� �2

ϕ2xϕ2xx

+ 4 f ′′� �2

ϕ3xϕxxx + f ′ f ′′ϕ2xϕxxxx + f ′′ϕ2xu0xx+ f ′ f 4ð Þϕ4xϕxx + 6f ′ f 3ð Þϕ2xϕ

2xx + 3f ′ f ′′ϕ3xx

+ 4f ′ f ′′ϕxϕxxϕxxx + f ′� �2

ϕxxϕxxxx

+ f ′ϕxxu0xx + f 4ð Þu0ϕ4x + 6f 3ð Þu0ϕ

2xϕxx

+ 3f ′′u0ϕ2xx + 4f ′′u0ϕxϕxxx + f ′u0ϕxxxx + u0u0xx ,ð10Þ

uxxxx = f 6ð Þϕ6x + 15f 5ð Þϕ4xϕxx + 45f 4ð Þϕ2xϕ2xx

+ 20f 4ð Þϕ3xϕxxx + 15f 3ð Þϕ3xx+ 60f 3ð Þϕxϕxxϕxxx + 15f 3ð Þϕ2xϕxxxx+ 10f ′′ϕ2xx + 15f ′′ϕxxϕxxxx+ 6f ′′ϕxϕxxxx + f ′ϕxxxxxx + u0xxxx:

ð11Þ

2 Advances in Mathematical Physics

Table 1: Information of equilibrium points.

No. Values of parameters detM P1ð Þ detM P2ð Þ P1 P2 Phase portraits

1 c > 0, a < 0 J P1ð Þ > 0 J P2ð Þ < 0 center point saddle point Figure 1(a)

c > 0, a > 0, ∣k2 ∣ >ffiffiffia

p∣ k1 ∣ J P1ð Þ > 0 J P2ð Þ < 0 center point saddle point Figure 1(b)

c < 0, a > 0, ∣k2 ∣ <ffiffiffia

p∣ k1 ∣ J P1ð Þ > 0 J P2ð Þ < 0 center point saddle point Figure 1(c)

c < 0, a < 0 J P1ð Þ < 0 J P2ð Þ > 0 saddle point center point Figure 2(a)

c > 0, a > 0, ∣k2 ∣ <ffiffiffia

p∣ k1 ∣ J P1ð Þ < 0 J P2ð Þ > 0 saddle point center point Figure 2(b)

c < 0, a > 0, ∣k2 ∣ >ffiffiffia

p∣ k1 ∣ J P1ð Þ < 0 J P2ð Þ > 0 saddle point center point Figure 2(c)

4

2

0

–2

–4

–1 1 2 3 4X

Y

(a)

4

2

0

–2

–4

–1 1 2 43 5X

Y

(b)

2

1

0

–1

–2

–3–4 –2 –1 1X

Y

(c)

Figure 1: Phase portraits of system (50).

3Advances in Mathematical Physics

6

4

2

0

–2

–4

–6

–2 –1

Y

X2 3–3 1

(a)

4

2

0

–2

–4

–2 –1 1 2 3 4 5

Y

X

(b)

Figure 2: Continued.


Now, substituting (7)–(11) into Equation (1) and sim-plifying, it yields

ϕ6x −2b f 3ð Þ� �2

− 2bf 4ð Þ f ′′ + cf 6ð Þ� �

+ ϕ4x −af 4ð Þ − 24bf ′′f 3ð Þ + 2bf ′ f 4ð Þ − 15cf 5ð Þ� �

ϕxx

h− 2bu0 f 4ð Þ

i+ ϕ3x −4bf ′ f 3ð Þ − 8b f ′

� �2��

+ 20cf 4ð Þ�ϕxxx − 4bf 3ð Þu0x� + ϕ2x f 4ð Þϕ2t + f 3ð Þϕtt

h− 6a + 12bu0ð Þf 3ð Þϕxx + −24b f ′′

� �2�

+ 2bf ′ f 3ð Þ + 45cf 4ð Þ�ϕ2xx

+ −2bf ′ f ′′ + 15cf 3ð Þ� �

ϕxxxx − 2bf ′′u0xxi

+ ϕx 4f 3ð Þϕtϕxt + 2f ′′ϕxtth

+ −4af ′′ − 20bf ′ f ′′ϕxx + 60cf 3ð Þϕxx − 8bu0 f ′′� �

ϕxxx

+ 6cf ′′ϕxxxx − 12bu0x f ′′ϕxxi,

ð12Þ

ϕ3xx −6bf ′ f ′′ + 15cf 3ð Þ� �

− 3a + 6bu0ð Þf ′′ϕ2xx

+ ϕxx f 3ð Þϕ2t − 2b f ′� �2

ϕxxxx + 15cf ′′ϕxxxx�

+ f ′′ϕtt − 2bf ′u0xxi+ ϕ2xxx −2b f ′

� �2+ 10cf ′′

� �− 4bu0x f ′ϕxxx − a + 2bu0ð Þf ′ϕxxxx + 2f ′′ϕ2xt+ f ′ϕttxx + 2f ′′ϕtϕxxt + cf ′ϕxxxxxx + u0tt − au0xx+ cu0xxxx − 2bu20x − 2bu0u0xx = 0:

ð13Þ

Setting the coefficient of the term ϕ6x in Equation (13)to zero, we obtain a nonlinear ordinary differential equa-tion for function f ðϕÞ:

−2b f 3ð Þ� �2

− 2bf 4ð Þ f ′′ + cf 6ð Þ = 0: ð14Þ

Integrating Equation (14) once and neglecting theconstant of integration gives

−2bf 3ð Þ f ′′ + cf 5ð Þ = 0: ð15Þ

Again integrating Equation (15) and letting the con-stant of integration be zero, we obtain

−b f ′′� �2

+ cf 4ð Þ = 0, ð16Þ

which has particular solution

f ϕð Þ = r ln ϕ, ð17Þ

where r = −6c/b:According to (17), we get the following results:

f ′� �2

= −rf ′′, f ′′f ′ = −r2 f

3ð Þ, f 3ð Þ f ′ = −r3 f

4ð Þ, ð18Þ

f 4ð Þ f ′ = −r4 f

5ð Þ, f ′′� �2

= −r6 f

4ð Þ, f ′′f 3ð Þ = −r12 f

5ð Þ:

ð19Þ

4

2

0–2 –1 1 2 3 4

–2

–4

Y

X

(c)

Figure 2: Phase portraits of system (50).


By (6) and (17), we obtain the auto-B€acklund transfor-mation of Equation (1) as follows:

u x, tð Þ = −rϕ2xϕ2

+ rϕxxϕ

+ u0: ð20Þ

Letting the seed solution

u0 x, tð Þ = 0, ð21Þ

and using results (19), then Equation (13) can be simplifiedas

f 4ð Þ −aϕ2x + 4cϕxϕxxx + ϕ2t − 3cϕ2xx�

ϕ2x

+ f 3ð Þ ϕtt − 6aϕxx + 9cϕxxxxð Þϕ2x�

+ 4ϕxϕtϕxt − 3cϕ3xx + ϕxxϕ2t

+ f ′′ ϕ2xt + 2ϕxϕxtt + ϕttϕxx + 2ϕtϕxxt

�− 3aϕ2xx − 4aϕxϕxxx + 6cϕxϕxxxxx− 2cϕ2xxx + 3cϕxxϕxxxx

+ f ′ ϕxxtt − aϕxxxx + cϕxxxxxx½ � = 0:

ð22Þ

0.04

0.03

0.02

0.01

–0.01

–0.02

–0.03

–0.04–0.015 –0.01 –0.005

g0 = 0, 𝜔 = 0.01

0X

Y

0.005 0.01 0.015

0

(a)

0.04

0.03

0.02

0.01

–0.01

–0.02

–0.03

–0.04–0.02 –0.01 0

X

0.01 0.02

Y 0

g0 = 0.01, 𝜔 = 0.01

(b)

0.02

g0 = 0.1, 𝜔 = 0.01

0.01

0

–0.01

–0.02

–0.025 –0.02 –0.015 –0.01 –0.005 0X

Y

(c)

3

2

1

0

–1

–2

–3–1.5 –1 –0.5

Y

0 0.5X

g0 = 5, 𝜔 = 0.01

(d)

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.


Setting the coefficients of f ð4Þ, f ð3Þ, f ′′, and f ′ in Equa-tion (22) to zero yields a set of partial differential equationsfor ϕðx, tÞ as follows:

ϕ2x −aϕ2x + 4cϕxϕxxx + ϕ2t − 3cϕ2xx �

= 0, ð23Þ

ϕtt − 6aϕxx + 9cϕxxxxð Þϕ2x + 4ϕxϕtϕxt − 3cϕ3xx + ϕxxϕ2t = 0,

ð24Þ

ϕ2xt + 2ϕxϕxtt + ϕttϕxx + 2ϕtϕxxt − 3aϕ2xx− 4aϕxϕxxx + 6cϕxϕxxxxx − 2cϕ2xxx + 3cϕxxϕxxxx = 0,

ð25Þ

ϕtt − aϕxx + cϕxxxxð Þxx = 0: ð26Þ

Here, Equations (24) and (25) can be rewritten as

ϕ2x ϕtt − aϕxx + cϕxxxx½ � + ϕxx −aϕ2x + 4cϕxϕxxx + ϕ2t − 3cϕ2xx�

+ 2ϕx −aϕ2x + 4cϕxϕxxx + ϕ2t − 3cϕ2xx�

x= 0,

ð27Þ

ϕtt ϕtt − aϕxx + cϕxxxx½ � + 2ϕx ϕtt − aϕxx + cϕxxxx½ �x+ −aϕ2x + 4cϕxϕxxx + ϕ2t − 3cϕ2xx�

xx= 0:

ð28Þ

By analysis of Equations (27) and (28), we find thatEquations (23)–(26) are satisfied automatical under the fol-lowing conditions:

ϕtt − aϕxx + cϕxxxx = 0, ð29Þ

−aϕ2x + 4cϕxϕxxx + ϕ2t − 3cϕ2xx = 0: ð30Þ

0.04g0 = 0, 𝜔 = 10

0.02

0

–0.02

–0.04–0.02 –0.01 0 0.01 0.02

X

Y

(a)

–0.02 –0.01 0 0.01 0.02X

0.04

0.02

0

–0.02

–0.04

Y

(b)

g0 = 0.1, 𝜔 = 10

–0.02 –0.01 0 0.01 0.02X

0.05

0

–0.05

Y

(c)

g0 = 10, 𝜔 = 10

–0.4 –0.2 0 0.2 0.4X

1.5

0.5

1

0

–0.5

–1

–1.5

Y

(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.


To obtain some exact solutions of Equations (29) and(30), we assume the solutions of the form:

ϕ x, tð Þ = A + B sin ξð Þ exp ηð Þ, ð31Þ

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 sim-plifying, it leads to a system of nonlinear algebraic equationswith respect to K , Ω, k,and ω as follows

−Ω2 + ω2 + aK2 − ak2 + cK4 − 6cK2k2 + ck4 = 0,2Ωω − 2aKk − 4ckK3 + 4cKk3 = 0,

−2aKk + 2Ωω + 12c − 4ð ÞkK3 − 12c − 4ð ÞKk3 = 0,−ak2 + ω2 + ck4 − 6cK2k2 + aK2 −Ω2 + cK4 = 0,

Ω2 − aK2 − 4cK4 = 0:

ð32Þ

By solving the above equations, the following sets ofsolutions are obtained

Case 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 =ffiffiffiffiffiffiffiffiffiffiffi−a/4c

p, ω = ±

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi3a2/16c

p, where c >

0, a < 0, c ≠ 1/3:

Case 3. k = 0,Ω = 0, K = −ffiffiffiffiffiffiffiffiffiffiffi−a/4c

p, ω = ±


p, where c

> 0, a < 0, c ≠ 1/3:

Case 4. Ω =Ω, k =ffiffiffiffiffiffiffiffiffiffiffiffiffiffi−Ω2/4c4

p, K = −

ffiffiffiffiffiffiffiffiffiffiffiffiffiffi−Ω2/4c4

p, ω = 2Ω, where

c < 0, c ≠ 1/3:

Case 5. Ω =Ω, k = −ffiffiffiffiffiffiffiffiffiffiffiffiffiffi−Ω2/4c4

p, K =


p, ω = 2Ω, where

c < 0, c ≠ 1/3:

0.03g0 = 0.1, 𝜔 = 0.1

0.02

0.01

–0.01

–0.02

–0.03–0.02 –0.01 0 0.01 0.02

X

Y 0

(a)

0.04

0.02

0

–0.02

–0.04–0.04 –0.02 0 0.02 0.04

g0 = 0.1, 𝜔 = 1

X

Y

(b)

0.05

–0.05

0

g0 = 0.1, 𝜔 = 10

–0.02 –0.01 0 0.01 0.02X

Y

(c)

0.04

0.02

–0.02

–0.04

0

g0 = 0.1, 𝜔 = 100

–0.02 –0.01 0 0.01 0.02X

Y

(d)

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.


Case 6. Ω =Ω, k =ffiffiffiffiffiffiffiffiffiffiffiffiffiffi3Ω2/4c4

p, K =

ffiffiffiffiffiffiffiffiffiffiffiffiffiffi3Ω2/4c4

p, ω = −2Ω, where

c > 0, c ≠ 1/3:

Case 7. Ω =Ω, k = −ffiffiffiffiffiffiffiffiffiffiffiffiffiffi3Ω2/4c4

p, K = −


p, ω = −2Ω,

where c > 0, c ≠ 1/3:

According to (20), (21), and (31), the solution of thefourth-order nonlinear generalized Boussinesq water waveequation is as follows:

u x, tð Þ = −rϕ2xϕ2

+ rϕxxϕ

= 6cB2 exp 2ηð Þ K cos ξð Þ + k sin ξð Þ½ �2b A + B sin ξð Þ exp ηð Þ½ �2

−6cB exp ηð Þ k2 − K2 �

sin ξð Þ + 2Kk cos ξð Þ� b A + B sin ξð Þ exp ηð Þ½ � ,

ð33Þ

where ξ = Kx +Ωt + ξ0, η = kx + ωt + η0:

For Case 1, the exact explicit solutions to the fourth-order nonlinear generalized Boussinesq water wave equationare obtained as follows:

u x, tð Þ = 6cB2k2 sin2ξ0 exp 2 kx + ωt + η0ð Þb A + B sin ξ0 exp 2 kx + ωt + η0ð Þ½ �2

−6cBk2 sin ξ0 exp kx + ωt + η0ð Þ

b A + B sin ξ0 exp 2 kx + ωt + η0ð Þ½ � :ð34Þ

If A = 1, B sin ξ0 = 1, Equation (34) is reduced to the sol-itary wave solutions as follows:

u x, tð Þ = −3ck22b sech2 1

2 kx + ωt + η0ð Þ� �

: ð35Þ


2

1

0

–1

–2–0.5 0 0.5 1 1.5

X

Y

g0 = 5, 𝜔 = 0.001

(a)

X

3

2

2

1

1

0

0

–1

–1–3

–2

–2

Y

g0 = 5, 𝜔 = 0.1

(b)

–5

0

5

X2 310–1–2

Y

g0 = 5, 𝜔 = 1

(c)

X

0 0.1–0.1–0.2

–0.5

0

0.5

1

–10.2

Y

g0 = 5, 𝜔 = 10

(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.


If A = 0, Equation (36) is reduced to

u x, tð Þ = −3a2b csc2

ffiffiffiffiffiffi−a4c

rx + ξ0

!, ð37Þ

which are periodic wave solutions.


If A = 0, Equation (38) is reduced to Equation (37). For Case 4, the exact explicit solutions to the fourth-order nonlinear generalized Boussinesq water wave equationare obtained as follows:


u x, tð Þ =−3aB2 exp ±


p� �t + η0

� �exp ±


p� �t + η0

� �+ A sin −

ffiffiffiffiffiffiffiffiffiffiffi−a/4c

p �x + ξ0

�h i2cb A + B sin −


px + ξ0

�exp ±


pt + η0

� �h i2 : ð38Þ

u x, tð Þ =−6cB


p� �exp


p� �x + 2Ωt + η0

� �cos −


p� �x +Ωt + ξ0

� �− sin −


p� �x +Ωt + ξ0

� �h ib A + B sin −


p� �x +Ωt + ξ0

� �exp


p� �x + 2Ωt + η0

� �h i2

+12cB

ffiffiffiffiffiffiffiffiffiffiffiffiffiffi−Ω2/4c

p� �exp


p� �x + 2Ωt + η0

� �cos −


p� �x +Ωt + ξ0

� �b A + B sin −


p� �x +Ωt + ξ0

� �exp


p� �x + 2Ωt + η0

� �h i :

ð39Þ

u x, tð Þ =6cB


pexp −


px + 2Ωt + η0

� �cos


px +Ωt + ξ0

� �− sin


px +Ωt + ξ0

� �h ib A + B sin


px +Ωt + ξ0

� �exp −


px + 2Ωt + η0

� �h i2

+12cB

ffiffiffiffiffiffiffiffiffiffiffiffiffiffi−Ω2/4c

pexp −


px + 2Ωt + η0

� �cos


px +Ωt + ξ0

� �b A + B sin


px +Ωt + ξ0

� �exp −


px + 2Ωt + η0

� �h i :

ð40Þ

u x, tð Þ =−3aB2 exp ±


p� �t + η0

� �exp ±


p� �t + η0

� �+ A sin


p �x + ξ0

�h i2cb A + B sin


p �x + ξ0

�exp ±


p� �t + η0

� �h i2 : ð36Þ




Remark 1. Equations (29) and (30) may also have otherforms of solutions as follows:

ϕ x, tð Þ = A + B cos ξð Þ exp ηð Þ,ϕ x, tð Þ = A + B sinh ξð Þ exp ηð Þ,ϕ x, tð Þ = A + B cosh ξð Þ exp ηð Þ,

ð43Þ

where ξ = Kx +Ωt + ξ0, η = kx + ωt + η0 ;A, B, K ,Ω, ξ0, k, ω, and η0 are constants which are to bedetermined.

3. Dynamical Behaviors and ChaoticBehaviors of Equation (1)

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

3.1. Bifurcation and Phase Portraits of Equation (1). In Ref.[9], Lie point symmetries admitted by Equation (1) weregiven as follows:

V1 =∂∂t

,V2 =∂∂x

,

V3 = 2bt ∂∂t

+ bx∂∂x

− a + 2buð Þ ∂∂u

,ð44Þ

which form a three-dimensional Lie algebra.We consider the Lie point symmetry as follows:

k1V1 + k2V2 = k1∂∂t

+ k2∂∂x

, ð45Þ

where k1, k2 are constants.The corresponding characteristic equation is

dtk1

= dxk2

= du0 : ð46Þ

Solving the above characteristic equation, we obtain thegroup invariant solution:

u x, tð Þ = X ξð Þ, ð47Þ

u x, tð Þ =6cB


pexp


px − 2Ωt + η0

� �cos


px +Ωt + ξ0

� �− sin


px +Ωt + ξ0

� �h ib A + B sin


px +Ωt + ξ0

� �exp


px − 2Ωt + η0

� �h i2

−12cB

ffiffiffiffiffiffiffiffiffiffiffiffiffiffi3Ω2/4c

pexp


px − 2Ωt + η0

� �cos


px +Ωt + ξ0

� �b A + Bsin


px +Ωt + ξ0

� �exp


px − 2Ωt + η0

� �h i :

ð41Þ

u x, tð Þ = −6cB


pexp −


px − 2Ωt + η0

� �cos −


px +Ωt + ξ0

� �− sin −


px +Ωt + ξ0

� �h ib A + B sin −


px +Ωt + ξ0

� �exp −


px − 2Ωt + η0

� �h i2

−12cB

ffiffiffiffiffiffiffiffiffiffiffiffiffiffi3Ω2/4c

pexp −


px − 2Ωt + η0

� �cos −


px +Ωt + ξ0

� �b A + B sin −


px +Ωt + ξ0

� �exp −


px − 2Ωt + η0

� �h i :

ð42Þ


where ξ = k1x − k2t is the similarity invariant. This is a trav-eling wave transformation, which we will use to analyzedynamic behaviors of the equation.

Under the traveling wave transformation, Equation (1) isreduced to

k22 − ak21 �

X ′′ − 2bk21 X ′� �2

+ XX ′′� �

+ ck41X4ð Þ = 0, ð48Þ

where the ‘‘′” denotes the differentiation with respect to thevariable ξ.

Integrating Equation (48) twice with respect to ξ andvanishing the constants of integration yields

X ′′ = αX2 + βX: ð49Þ

where α = b/ck21, β = ðak21 − k22Þ/ck41:Letting X ′ = Y , Equation (49) is equivalent to the two-

dimensional plane autonomous system:

dXdξ

= Y ,

dYdξ

= αX2 + βX:

0BBB@ ð50Þ

Obviously, system (50) is an integrable Hamiltonian sys-tem. Applying the first integral, we get Hamiltonian func-tions as follows:

12Y

2 −α

3 X3 −

β

2 X2 = h, ð51Þ

where h is the constant of integration.By solving the following system:

Y = 0,αX2 + βX = 0,

ð52Þ

equilibrium points of system (52) can be obtained as P1ð0, 0Þand P2ð−β/α, 0Þ.

Let MðPiÞ denote the coefficient matrix of the linearizedsystem of system (50) at equilibrium points PiðXi, 0Þði = 1,2Þ as follows:

M Pið Þ =0 1

2αXi + β 0

!, ð53Þ

where the trace ofMðPiÞ is zero and the determinant ofM isJðPiÞ = det MðPiÞ = −ð2αXi + βÞ. Then, we obtain det MðP1Þ = −β and det MðP2Þ = β.

By the theory of planar dynamical systems [24–30], wedraw the following conclusion as in Table 1. The differentphase portraits of system (50) are shown in Figures 1 and 2.

3.2. Chaotic Behaviors of Equation (1). In this section, thedynamics of system (50) perturbed by periodic term is to

be investigated via numerical simulations. For this purpose,suppose that periodic term is g0 cos ðωξÞ. Letting Z = ωξ,system (50) is modified as following three-dimensional sys-tem:

dXdξ

= Y ,

dYdξ

= αX2 + BβX + g0 cos Zð Þ,

dZdξ

= ω:

0BBBBBBBB@

ð54Þ

In the simulations, choosing parameters α = 2, β = −8,the effects of amplitude and frequency on the dynamics ofsystem (54) are to be discussed.

Firstly, the effect of amplitude is considered. For thisend, frequency ω is fixed while g0 is chosen as differentvalues. The corresponding phase portraits are depicted inFigures 3 and 4. By analyzing Figures 3 and 4, we can getthe following results.

Case 1. When g0 = 0. System (54) is undisturbed by periodicsignal, and it is provided with periodic solution (seeFigures 3(a) and 4(a)).

Case 2. When g0 ≠ 0. Figure 3 means that the system (54) isdisturbed by periodic signal, and system (54) presents limitcycles for low frequency (see Figures 3(b)–3(d)). Figure 4shows that, when system (54) is disturbed by periodic signalwith higher frequency, it can appear as limit cycles (seeFigures 4(b) and 4(c)) and chaotic phenomenon (seeFigure 4(d))with the change of amplitude.

Consequently, Figures 3 and 4 indicate that the ampli-tude has much effect on the dynamics of system (54).

Secondly, g0 taken as a constant, and ω is chosen as dif-ferent values. The corresponding phase portraits are calcu-lated and shown in Figures 5 and 6. In Figure 5, it isobvious to see that, for small amplitude g0 = 0:1, with thechange 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 smallamplitude g0 = 5, system (54) mainly shows chaos.

4. Conclusions

The extended homogeneous balance method is an effectivemethod solving nonlinear partial differential equations. Weapplied it to obtain the auto-B€acklund transformation trans-formation and some new exact explicit solutions for thefourth-order nonlinear generalized Boussinesq water waveequation. Using the theory of plane dynamic system, thedynamical behavior analysis of the perturbed dynamic sys-tem and the chaotic behaviors of the perturbed system areanalyzed. The change of the physical parameters will affectthe dynamic behavior of the dynamic system.


Data Availability

There is no database involved in the manuscript.

Conflicts of Interest

The authors declare no potential conflict of interest.

Acknowledgments

This work is supported by National Natural Science Founda-tion of China (Grant Nos. 51777180, 12075190) and by Nat-ural Science Foundation of Shaanxi Province (Grant No2021JM-455).

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