The chaotic behavior and traveling wave solutions of the conformable extended Korteweg–de-Vries model

Chunyan Liu

doi:10.1515/phys-2024-0069

Article Open Access

The chaotic behavior and traveling wave solutions of the conformable extended Korteweg–de-Vries model

Chunyan Liu

Published/Copyright: August 19, 2024

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From the journal Open Physics Volume 22 Issue 1

Abstract

In this article, the phase portraits, chaotic patterns, and traveling wave solutions of the conformable extended Korteweg–de-Vries (KdV) model are investigated. First, the conformal fractional order extended KdV model is transformed into ordinary differential equation through traveling wave transformation. Second, two-dimensional (2D) planar dynamical system is presented and its chaotic behavior is studied by using the planar dynamical system method. Moreover, some three-dimensional (3D), 2D phase portraits and the Lyapunov exponent diagram are drawn. Finally, many meaningful solutions are constructed by using the complete discriminant system method, which include rational, trigonometric, hyperbolic, and Jacobi elliptic function solutions. In order to facilitate readers to see the impact of fractional order changes more intuitively, Maple software is used to draw 2D graphics, 3D graphics, density plots, contour plots, and comparison charts of some obtained solutions.

Keywords: Korteweg–de-Vries model; phase portrait; complete discriminant system; traveling wave solution

1 Introduction

Nonlinear partial differential equations [1–10] are of great significance in our study of nonlinear problems, but their solution is very difficult, and it is even more difficult to obtain traveling wave solutions [11–22]. Generally, it is only possible to obtain some traveling wave solutions under very special conditions, and even if they are obtained, there are not unified methods. In recent years, through the efforts of mathematicians and physicists in research and experiments, they have been found that there are some effective methods for constructing traveling wave solutions. For example, Bäcklund transformation, Darboux transformation, homogeneous equilibrium method, Riccati equation method, F-expansion method, variable separation method, algebraic geometry method, various function expansion methods, auxiliary function method, polynomial complete discriminant system method, etc. [23–32]. With the emergence of various solving methods, not only difficult to solve equations in the past have been solved, but new physically significant solutions have also been continuously discovered and applied.

In recent years, many scholars have begun to pay attention to fractional differential equations. They found that fractional derivative models [33–39] have more advantages than integer derivative models, such as fractional calculus operators being non-local quasi-differential operators, making them of great significance in the study of nonlinear fields. Moreover, fractional derivatives and fractional integrals are widely used in fields such as physics, biology, and chemistry. Compared with linear differential equations, fractional order differential equations do not have unified and effective solutions [40–44]. Many scholars have begun to study the exact solutions of fractional order differential equations [45–51], they used different methods to obtain many new traveling wave solutions. Due to the complexity of fractional partial differential equations, we can try to use different methods to handle them, and the results obtained will also vary.

Due to the importance of Korteweg–de-Vries (KdV) equation in nonlinear physics, many scholars have studied the KdV equation [52–57]. Recently, Eslami [58] has obtained the conformable extended KdV equation by using the reductive perturbation approach as follows:

(1.1) D τ ϱ φ + λ 3 2 3 2 λ 4 − ( 1 − f ) C 2 − f σ 2 D 2 D ξ 2 ϱ φ + λ 3 2 5 2 λ 6 − ( 1 − f ) C 3 − f σ 3 D 3 ( D ξ ϱ φ ) 3 + λ 3 2 D ξ 3 ϱ φ = 0 ,

where φ is a function of τ and ξ , τ = ε 3 t , ξ = ε ( x − λ t ) , C 2 , C 3 , D 2 , D 3 , f , σ , ε are parameters, and λ is the phase velocity of solitons. The definition and properties of D τ ϱ φ can be found in the study of Li and Han [59]. In this article, we use the complete discriminant system of polynomials to study the traveling wave solutions of Eq. (1.1).

This article is organized as follows: In Section 2, we provide the definition of fractional derivatives and perform a traveling wave transformation on Eq. (1.1). In Section 3, we present the two-dimensional (2D) planar dynamical system and its chaotic patterns by using the planar dynamical system method, use Maple software to draw some three-dimensional (3D) and 2D phase portraits, and use Matlab software to plot the Lyapunov exponent diagram for the conformable extended KdV model. In Section 4, we use the complete discriminant system method of quartic polynomials to find the traveling wave solutions of Eq. (1.1). In Section 5, we conduct numerical analysis and use Maple software to draw 2D graphics, 3D graphics, density plots, contour plots, and comparison chart of some solutions. Finally, we summarize the results in Section 6.

2 Mathematical preliminaries

In this section, we review the definition of conformable fractional derivatives.

Definition 2.1

[59] Let f : [ 0 , ∞ ) → R . Then, the conformable fractional derivative of f of order α is defined as

(2.1) D t ϱ f ( t ) = lim ε → 0 f ( t + ε t 1 − ϱ ) − f ( t ) ε , ∀ t ∈ [ 0 , + ∞ ) , ϱ ∈ ( 0 , 1 ] ,

the function f is ϱ -conformable differentiable at a point t if the limit in Eq. (2.1) exists.

Proposition 2.2

[59] The conformable fractional derivative possesses the following properties:

D t ϱ ( t μ ) = μ t μ − ϱ , ∀ μ ∈ R .
D t ϱ ( a f ( t ) + b g ( t ) ) = a D t ϱ f ( t ) + b D t ϱ g ( t ) , ∀ a , b ∈ R .
D t ϱ ( f ∘ g ) ( t ) = t 1 − ϱ g ( t ) ϱ − 1 g ′ ( t ) D t ϱ ( f ( t ) ) ∣ t = g ( t ) .

Next, we assume φ ( τ , ξ ) = ϕ ( χ ) , χ = ( 1 ϱ ) ξ ϱ − V ( 1 ϱ ) τ ϱ , then Eq. (1.1) can be transformed into

(2.2) − V ϕ + λ 3 2 3 2 λ 4 − ( 1 − f ) C 2 − f σ 2 D 2 ϕ 2 + λ 3 2 5 2 λ 6 − ( 1 − f ) C 3 − f σ 3 D 3 ϕ 3 + λ 3 2 ϕ ″ = 0 .

By integrating Eq. (2.2) once, we obtain

(2.3) ( ϕ ′ ) 2 = a 4 ϕ 4 + a 3 ϕ 3 + a 2 ϕ 2 + a 1 ϕ + a 0 ,

where a 4 = − 1 2 5 2 λ 6 − ( 1 − f ) C 3 − f σ 3 D 3 , a 3 = − 2 3 3 2 λ 4 − ( 1 − f ) C 2 − f σ 2 D 2 , a 2 = 2 V λ 3 , a 1 = 0 , a 0 is any integral constant.

When a 4 > 0 , we make the following transformation:

(2.4) ω = ( a 4 ) 1 4 ϕ + a 3 4 a 4 , η = ( a 4 ) 1 4 χ ,

where ϕ is the function of χ and ω is the function of η .

Substituting Eq. (2.4) into Eq. (2.3) yields

(2.5) ( ω ′ ) 2 = F ( ω ) = ω 4 + p ω 2 + q ω + r ,

where p = ( a 2 − 3 a 3 2 8 a 4 ) ( a 4 ) − 1 2 , q = ( a 3 3 8 a 4 2 − a 2 a 3 2 a 4 + a 1 ) ( a 4 ) − 1 4 , r = − 3 a 3 4 256 a 4 3 + a 2 a 3 2 16 a 4 2 − a 1 a 3 4 a 4 + a 0 .

When a 4 < 0 , we make the following transformation:

(2.6) ω = ( − a 4 ) 1 4 ϕ + a 3 4 a 4 , η = ( − a 4 ) 1 4 χ ,

where ϕ is the function of χ and ω is the function of η .

Substituting Eq. (2.6) into Eq. (2.3) yields

(2.7) ( ω ′ ) 2 = − F ( ω ) = − ( ω 4 + p ω 2 + q ω + r ) ,

where p = ( − a 2 + 3 a 3 2 8 a 4 ) ( − a 4 ) − 1 2 , q = ( − a 3 3 8 a 4 2 + a 2 a 3 2 a 4 − a 1 ) ( − a 4 ) − 1 4 , and r = 3 a 3 4 256 a 4 3 − a 2 a 3 2 16 a 4 2 + a 1 a 3 4 a 4 − a 0 .

The complete discriminant system of the quartic polynomial F ( ω ) = ω 4 + p ω 2 + q ω + r is as follows: G 1 = 4 , G 2 = − p , G 3 = − 2 p 3 + 8 p r − 9 q 2 , G 4 = − p 3 q 2 + 4 p 4 r + 36 p q 2 r − 32 p 2 r 2 − 27 4 q 4 + 64 r 3 , E 2 = 9 q 2 − 32 p r .

3 Phase portraits

In this section, we will write Eq. (2.2) in the following form:

(3.1) ϕ ″ + A 3 ϕ 3 + A 2 ϕ 2 + A 1 ϕ = 0 ,

where A 1 = − 2 V λ 3 , A 2 = 3 2 λ 4 − ( 1 − f ) C 2 − f σ 2 D 2 , A 3 = 5 2 λ 6 − ( 1 − f ) C 3 − f σ 3 D 3 . We conduct dynamic behavior analysis on Eq. (3.1). Supposing ϕ ′ = z , Eq. (3.1) can be represented by the following planar dynamic system [60]:

(3.2) d ϕ d χ = z , d z d χ = − A 3 ϕ 3 − A 2 ϕ 2 − A 1 ϕ .

Thus, we can obtain the phase portraits with arrows of Eq. (3.2) in Figure 1.

$Figure 1 The phase portraits with arrows of Eq. (3.2). (a) A 1 = ‒ 2 {A}_{1}=‒2 , A 2 = 1 2 {A}_{2}=\frac{1}{2} , and A 3 = 1 2 {A}_{3}=\frac{1}{2} and (b) A 1 = 2 {A}_{1}=2 , A 2 = 1 2 {A}_{2}=\frac{1}{2} , and A 3 = 1 2 {A}_{3}=\frac{1}{2} .$

Figure 1

The phase portraits with arrows of Eq. (3.2). (a) A 1 = ‒ 2 , A 2 = 1 2 , and A 3 = 1 2 and (b) A 1 = 2 , A 2 = 1 2 , and A 3 = 1 2 .

In order to study the chaotic behavior caused by periodic disturbances, we introduce the disturbance factor such as A 0 cos ( ω χ ) , where A 0 represents amplitude and ω represents frequency. Next, we can use the following system of equations to represent the planar dynamical system:

(3.3) d ϕ d χ = z , d z d χ = − A 3 ϕ 3 − A 2 ϕ 2 − A 1 ϕ + A 0 cos y , d y d χ = ω ,

where y = ω χ . We can obtain the phase portraits of Eq. (3.3) in Figure 2 and the Lyapunov exponent diagram of Eq. (3.3) in Figure 3.

Figure 2

The phase portraits of Eq. (3.3). (a) 3D phase portrait with A 0 = 3 5 , A 1 = ‒ 2 , A 2 = 1 2 , A 3 = 1 2 , and ω = 5 4 , (b) 3D phase portrait with A 0 = 3 5 , A 1 = 2 , A 2 = 1 2 , A 3 = 1 2 , and ω = 5 4 , (c) 2D phase portrait with A 0 = 3 5 , A 1 = ‒ 2 , A 2 = 1 2 , A 3 = 1 2 , and ω = 5 4 , (d) 2D phase portrait with A 0 = 3 5 , A 1 = 2 , A 2 = 1 2 , A 3 = 1 2 , and ω = 5 4 .

$Figure 3 The Lyapunov exponent diagram of Eq. (3.3) with A 0 = 3 5 {A}_{0}=\frac{3}{5} , A 1 = ‒ 2 {A}_{1}=‒2 , A 2 = 1 2 {A}_{2}=\frac{1}{2} , A 3 = 1 2 {A}_{3}=\frac{1}{2} , and ω = 5 4 \omega =\frac{5}{4} .$

Figure 3

The Lyapunov exponent diagram of Eq. (3.3) with A 0 = 3 5 , A 1 = ‒ 2 , A 2 = 1 2 , A 3 = 1 2 , and ω = 5 4 .

Remark 3.1

Through the 3D and 2D phase portraits of system Eq. (3.3) in Figure 2, we can see that Figure 2(a) and (c) are chaotic behaviors. However, Figure 2(b) and (d) exhibit a quasi-periodic behavior.

Remark 3.2

Through the Lyapunov exponent diagram of system Eq. (3.3) in Figure 3, we can see that among the three variables, the exponent of variable ϕ tends to be 0.128 > 0 , the exponent of variable z tends to be 0, and the exponent of variable y tends to be − 0.128 . Since one of the variables has a Lyapunov exponent diagram greater than zero, we can explain that system Eq. (3.3) is a chaotic system.

4 Traveling wave solutions of Eq. (1.1)

First, we will write Eqs (2.5) and (2.6) in integral form as follows:

(4.1) ± ( η − η 0 ) = ∫ d ω κ ( ω 4 + p ω 2 + q ω + r ) ,

where η 0 is the integral constant and κ = ± 1 .

According to the complete discriminant system of polynomial [61] F ( ω ) = ω 4 + p ω 2 + q ω + r , the classification of all solutions of Eq. (4.1) can be obtained. There are nine cases in which the traveling wave solutions of Eq. (1.1) can be obtained.

Case 1. Suppose that G 2 < 0 , G 3 = 0 , G 4 = 0 , F ( ω ) = [ ( ω − a ) 2 + b 2 ] 2 , where a and b are real numbers and b > 0 . Let κ = 1 , it can be obtained from Eq. (4.1)

(4.2) η − η 0 = ∫ d ω ( ω − a ) 2 + b 2 = 1 b arctan ω − a b ,

where η 0 is the integral constant. So, the solution to Eq. (2.5) is

(4.3) ω = b tan [ b ( η − η 0 ) ] + a ,

it is a periodic solution of a trigonometric function. So, the solution of Eq. (1.1) is

(4.4) φ 1 = b tan b − 1 2 5 2 λ 6 − ( 1 − f ) C 3 − f σ 3 D 3 1 4 1 ϱ ξ ϱ − V 1 ϱ τ ϱ − η 0 + a × − 1 2 5 2 λ 6 − ( 1 − f ) C 3 − f σ 3 D 3 − 1 4 − 3 2 λ 4 − ( 1 − f ) C 2 − f σ 2 D 2 3 5 2 λ 6 − ( 1 − f ) C 3 − f σ 3 D 3 .

Case 2. Suppose that G 2 = 0 , G 3 = 0 , G 4 = 0 , and F ( ω ) = ω 4 . Let κ = 1 , it can be obtained from Eq. (4.1)

(4.5) η − η 0 = ∫ d ω ω 2 = − ω − 1 ,

where η 0 is the integral constant. So, the solution to Eq. (2.5) is

(4.6) ω = − ( η − η 0 ) − 1 ,

it is a rational number solution. So, the solution of Eq. (1.1) is

(4.7) φ 2 = − 1 2 5 2 λ 6 − ( 1 − f ) C 3 − f σ 3 D 3 1 4 × 1 ϱ ξ ϱ − V 1 ϱ τ ϱ − η 0 − 1 × − 1 2 5 2 λ 6 − ( 1 − f ) C 3 − f σ 3 D 3 − 1 4 − 3 2 λ 4 − ( 1 − f ) C 2 − f σ 2 D 2 3 5 2 λ 6 − ( 1 − f ) C 3 − f σ 3 D 3 .

Case 3. Suppose that G 2 > 0 , G 3 = 0 , G 4 = 0 , E 2 > 0 , F ( ω ) = ( ω − a ) 2 ( ω − b ) 2 , where a , b are real numbers and a > b . If κ = 1 , it can be obtained from Eq. (4.1)

(4.8) ± ( η − η 0 ) = ∫ d ω ( ω − a ) ( ω − b ) = 1 a − b ln ω − a ω − b ,

where η 0 is the integral constant.

(i) If ω > a or ω < b , the solution to Eq. (2.5) is

(4.9) ω = b − a e ( a − b ) ( η − η 0 ) − 1 + b = b − a 2 × coth ( a − b ) ( η − η 0 ) 2 − 1 + b .