New soliton solutions of the conformable time fractional Drinfel'd–Sokolov–Wilson equation based on the complete discriminant system method

Da Shi; Zhao Li

doi:10.1515/phys-2024-0099

Article Open Access

New soliton solutions of the conformable time fractional Drinfel'd–Sokolov–Wilson equation based on the complete discriminant system method

Da Shi and Zhao Li

Published/Copyright: November 25, 2024

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From the journal Open Physics Volume 22 Issue 1

Abstract

In this article, we mainly study new soliton solutions of the conformable time fractional Drinfel’d–Sokolov–Wilson (DSW) equation which has applications in a wide range of practical applications, including fluid dynamics problems. After fractional order travelling transformation, the partial differential equation studied in this article is transformed into ordinary differential equation and connected with a quadratic polynomial. By using the complete discriminant system of quadratic polynomial, we have achieved the classification of soliton solutions of the DSW equation and provided the precise expressions of their solutions, including hyperbolic function solutions, triangle function solutions and Jacobian elliptic function solutions. Some solutions obtained in this article are shown with 3D and 2D plots in order to investigate the propagation characteristics of soliton waves satisfying the conformable time fractional DSW equation. This method is not only fast and effective, but also can obtain new forms of solutions for DSW equation, which is beneficial for people to further understand fluid dynamics problems. This method is also applicable to studying the solutions of other types of partial differential equations.

Keywords: Drinfel’d–Sokolov–Wilson equation; fractional order complex transformation; soliton solution; complete discriminant system

1 Introduction

Nonlinear partial differential equations have a wide range of applications in practice [1–15]. With the continuous deepening of human understanding of the nonlinear characteristics of nature, a large number of new nonlinear evolution equations have emerged and become a research hotspot for studying numerous physical problems. For example, in the research of fluid mechanics, oceanography, optical fiber, communication systems, mathematical physics and other disciplines, people found a large number of dynamic behaviors under kink, cusp wave, bell-shaped, multi-wave, kink block, periodic block and kink wave modes. These physical phenomena belong to nonlinear evolution equation models. In this article, we will study new soliton solutions of the following conformable time fractional Drinfel’d–Sokolov–Wilson (DSW) equation [16]

(1.1) D t τ u + λ v v x = 0 , 0 < τ ⩽ 1 , D t τ v + ε u v x + μ u x v + ρ v x x x = 0 ,

where u = u ( x , t ) and v = v ( x , t ) are unknown functions. λ , ε , ρ and μ are all non-zero wave modes. The equation is mainly used to simulate nonlinear waves. In the study by Habibul et al. [16], obtained the soliton solutions of Eq. (1.1) by using the modified extended tanh method. However, further research is still needed on this equation, and more general Jacobian function solutions have not yet been obtained.

The solution of nonlinear equations is an important factor affecting their practical application. Scholars have conducted extensive research on this issue. Khater et al. [17,18] applied the extended simple equation method and generalized exp-expansion method to study the analytical solitary wave solutions of nonlinear partial differential equations. Gu and Yuan [19,20] proposed the generalized complex method to construct the closed form solutions of nonlinear fractional differential equations. Seadawy et al. [21,22] applied the extended direct algebraic method and Hirota bilinear technique to explore new solutions for partial differential equations. In previous studies [23–26], the authors explored the solutions for nonlinear partial differential equations using the generalized ( G ′ ⁄ G ) expansion method. Shafiqul et al. [27] proposed an improved F-expansion method combined with the Riccati equation to solve the exact solution of nonlinear evolution equations. Rezwan Ahamed et al. [28] studied the sine Gordon expansion method for solving traveling wave solutions of nonlinear equations. Mohyud-Din et al. [29] applied the fractional sub-equation method to obtain analytical solutions for two types of space–time fractional partial differential equations. Liu et al. [30–32] also studied the solutions of space–time fractional partial differential equations through methods such as symmetry analysis.

So far, scholars have conducted extensive research on the conditions and solutions of the DSW equation. In previous studies [33–36], the authors explored the energy conservation laws and symmetry of the equation. Multiple transformations are used to explore the abundant new exact solutions of the DSW equation, such as the modified extended tanh–coth method [37,38], algebraic method [39–41], improved F expansion method [42], Darboux transformation [43], improved Jacobi elliptic function method [44], trail function method [45], new auxiliary equation method [46], ( G ′ ⁄ G ) -expansion method [47,48], consistent Riccati expansion method [49], and the generalized bifurcation method [50]. In previous studies [51–54], the authors studied the approximate solution of the DSW equation. In previous studies [55–58], the authors conducted in-depth research on fractional derivatives [59–61] and conformable fractional derivatives.

In this study, we closely associate the solutions of Eq. (1.1) with the roots of a quadratic polynomial through transformations, and use the complete discriminant system of polynomials to classify the solutions of Eq. (1.1) and provide accurate expressions for each soliton solution. In this article, Section 2 briefly introduces the quadratic polynomial and its complete discriminant system used in the process of solving the DSW equation. Section 3 shows the detailed process of solving the exact solution of the conformable time fractional DSW equation based on the polynomial discriminant system. Section 4 provides a graphical representation of some solutions obtained in this article, with the reasonable assignment of parameters. Finally, we present the results of our research in Section 5.

2 Mathematical preliminaries

This article investigates the solutions of coupled nonlinear partial differential equations as follows:

(2.1) L ( u , v , u x , v x , u t , v t , u x x , v x x , u x x x , v x x x , D t τ u , D t τ v , … ) = 0 M ( u , v , u x , v x , u t , v t , u x x , v x x , u x x x , v x x x , D t τ u , D t τ v , … ) = 0 ,

where u = u ( x , t ) , v = v ( x , t ) , L and M are polynomials about functions u , v , and their derivatives.

Using the following fractional transformation [62] on Eq. (2.1)

(2.2) u ( x , t ) = U ( ξ ) , v ( x , t ) = V ( ξ ) , ξ = k x − θ t τ τ ,

where k , θ are nonzero constants, and 0 < τ ⩽ 1 , then Eq. (2.1) can be converted to the following ordinary differential equations:

(2.3) P ( U , V , U ′ , V ′ , U ″ , V ″ , U ‴ , V ‴ , … ) = 0 Q ( U , V , U ′ , V ′ , U ″ , V ″ , U ‴ , V ‴ , … ) = 0 ,

where U = U ( ξ ) , V = V ( ξ ) , P , and Q are polynomials about functions U , V , and their derivatives.

From Eq. (2.3), we can obtain

(2.4) H ( V , V ′ , V ″ , … ) = 0 ,

where H is a polynomial of V and its derivatives with respect to ξ .

If Eq. (2.4) can be transformed into the following equation:

(2.5) ( V ′ ) 2 = b 4 V 4 + b 2 V 2 + b 0 .

From the study of Liu [63], we can make the following assumptions

(2.6) V = ± ( 4 b 4 ) − 1 3 w , β 1 = 4 b 2 ( 4 b 4 ) − 2 3 , β 0 = 4 b 0 ( 4 b 4 ) − 1 3 , ζ = ( 4 b 4 ) 1 3 ξ ,

then, Eq. (2.4) can be converted to

(2.7) ( w ζ ′ ) 2 = w ( w 2 + β 1 w + β 0 ) .

Integrating Eq. (2.7), then we can obtain

(2.8) ± ( ζ − ζ 0 ) = ∫ d w w F ( w ) ,

where ζ 0 is an integral constant, F ( w ) = w 2 + β 1 w + β 0 .

The complete discriminant system of quadratic polynomial F ( w ) is as follows:

(2.9) Δ = β 1 2 − 4 β 0 .

According to the complete discriminant system (2.9), the classification of the roots of quadratic polynomial are as follows (see the study of Liu [63]):

Case 1: Δ = 0 .

When β 1 < 0 , the solution of Eq. (2.7) is

(2.10) w = − β 1 2 tanh 2 1 2 − β 1 2 ( ζ − ζ 0 ) ,

(2.11) w = − β 1 2 coth 2 1 2 − β 1 2 ( ζ − ζ 0 ) .

When β 1 > 0 , the solution of Eq. (2.7) is

(2.12) w = β 1 2 tan 2 1 2 β 1 2 ( ζ − ζ 0 ) .

When β 1 = 0 , the solution of Eq. (2.7) is

(2.13) w = β 1 2 tan 2 1 2 β 1 2 ( ζ − ζ 0 ) .

Case 2: Δ > 0 , β 0 = 0 .

When w > − β 1 , if β 1 > 0 , then the explicit solutions of Eq. (2.7) are

(2.14) w = β 1 2 tanh 2 1 2 β 1 2 ( ζ − ζ 0 ) − β 1 ,

(2.15) w = β 1 2 coth 2 1 2 β 1 2 ( ζ − ζ 0 ) − β 1 ,

When β 1 < 0 , then the explicit solution of Eq. (2.7) is

(2.16) w = − β 1 2 tanh 2 1 2 − β 1 2 ( ζ − ζ 0 ) − β 1 .

Case 3: Δ > 0 , β 0 ≠ 0 .

When β 1 > 0 , then the explicit solutions of Eq. (2.7) are

(2.17) w = β 1 2 tanh 2 1 2 β 1 2 ( ζ − ζ 0 ) − β 1 ,

(2.18) w = β 1 2 coth 2 1 2 β 1 2 ( ζ − ζ 0 ) − β 1 .

When w > c , the corresponding solution of Eq. (2.7) is

(2.19) w = − e sn 2 c − d ( ς − ς 0 ) 2 , m + c cn 2 c − d ( ς − ς 0 ) 2 , m ,

where m 2 = e − d c − d .

Case 4: Δ < 0 .

When w > 0 , then the solution of Eq. (2.7) is

(2.20) w = 2 β 0 1 + cn β 0 1 4 ( ς − ς 0 ) , m − β 0 ,

where m 2 = 1 2 1 − β 1 2 β 0 .

3 Applications

Applying the fractional composite transformation (2.2) on Eq. (1.1), then Eq. (1.1) can be converted to

(3.1) − k θ U ′ + λ k V V ′ = 0 , − k θ V ′ + ε k U V ′ + μ k U ′ V + ρ k 3 V ‴ = 0 .

From Eq. (3.1), we can derive the following result:

(3.2) − 6 θ 2 V + λ ( ε + 2 μ ) V 3 + 6 ρ θ k 2 V ″ = 0 ,

and the corresponding function U satisfies U = λ 2 θ V 2 .

Then

(3.3) V ″ = − λ ( ε + 2 μ ) 6 ρ θ k 2 V 3 + θ ρ k 2 V .

Integrating Eq. (3.3), we can obtain

(3.4) ( V ′ ) 2 = b 4 V 4 + b 2 V 2 + b 0 ,

where

(3.5) b 4 = − λ ( ε + 2 μ ) 12 ρ θ k 2 , b 2 = θ ρ k 2 ,

and b 0 is an arbitrary constant.

According to the discrimination system (2.9), the roots of F ( w ) will be classified, so correspondingly, we can obtain all exact solutions of Eq. (2.7).

Case 1: Δ = 0 .

When w > 0 , then Eq. (2.8) is transformed to

(3.6) ± ( ζ − ζ 0 ) = ∫ d w w F ( w ) = ∫ d w w + β 1 2 w ,

if β 1 < 0 , then Eq. (3.6) is converted to

(3.7) ± ( ζ − ζ 0 ) = − 2 β 1 ln 2 w − − β 1 2 w + − β 1 .

So, we can obtain the solution of Eq. (2.7)

(3.8) w = − β 1 2 tanh 2 1 2 − β 1 2 ( ζ − ζ 0 ) ,

(3.9) w = − β 1 2 coth 2 1 2 − β 1 2 ( ζ − ζ 0 ) .

Inserting Eq. (3.5) and Eq. (3.8) into Eq. (2.6), we can obtain

(3.10) V = ± 6 θ 2 λ ( ε + 2 μ ) tanh 2 × 1 2 − 7 2 1 3 θ 5 3 ρ 1 3 ( k λ ε + 2 μ k λ ) 2 3 − λ ε + 2 λ μ 3 ρ θ k 2 1 3 ξ − ς 0 1 2 .

Substituting Eq. (3.10) into Eq. (2.2), then the solution of Eq. (1.1) can be described as

(3.11) u 1 ( x , t ) = 3 θ ε + 2 μ tanh 2 1 2 − 7 2 1 3 θ 5 3 ρ 1 3 ( k λ ε + 2 μ k λ ) 2 3 − λ ε + 2 λ μ 3 ρ θ k 2 1 3 k x − θ t τ τ − ς 0 , v 1 ( x , t ) = ± 6 θ 2 λ ( ε + 2 μ ) tanh 2 1 2 − 7 2 1 3 θ 5 3 ρ 1 3 ( k λ ε + 2 μ k λ ) 2 3 − λ ε + 2 λ μ 3 ρ θ k 2 1 3 k x − θ t τ τ − ς 0 1 2 .

Based on Eq. (3.9), the solution of Eq. (1.1) can be described as

(3.12) u 2 ( x , t ) = 3 θ ε + 2 μ coth 2 1 2 − 7 2 1 3 θ 5 3 ρ 1 3 ( k λ ε + 2 μ k λ ) 2 3 − λ ε + 2 λ μ 3 ρ θ k 2 1 3 k x − θ t τ τ − ς 0 , v 2 ( x , t ) = ± 6 θ 2 λ ( ε + 2 μ ) coth 2 1 2 − 7 2 1 3 θ 5 3 ρ 1 3 ( k λ ε + 2 μ k λ ) 2 3 − λ ε + 2 λ μ 3 ρ θ k 2 1 3 k x − θ t τ τ − ς 0 1 2 .

If β 1 > 0 , we can obtain the solution of Eq. (2.7)

(3.13) w = β 1 2 tan 2 1 2 β 1 2 ( ζ − ζ 0 ) ,

then, the corresponding solution of Eq. (1.1) will be depicted as

(3.14) u 3 ( x , t ) = 3 θ ε + 2 μ tan 2 1 2 7 2 1 3 θ 5 3 ρ 1 3 ( k λ ε + 2 μ k λ ) 2 3 − λ ε + 2 λ μ 3 ρ θ k 2 1 3 k x − θ t τ τ − ς 0 , v 3 ( x , t ) = ± 6 θ 2 λ ( ε + 2 μ ) tan 2 1 2 7 2 1 3 θ 5 3 ρ 1 3 ( k λ ε + 2 μ k λ ) 2 3 − λ ε + 2 λ μ 3 ρ θ k 2 1 3 k x − θ t τ τ − ς 0 1 2 .

If β 1 = 0 , we can obtain the solution of Eq. (2.7)

(3.15) w = 4 ( ζ − ζ 0 ) 2 ,

then, the corresponding solution of Eq. (1.1) can be depicted as

(3.16) u 4 ( x , t ) = − 1 2 λ θ 2 3 3 ρ k 2 ε + 2 μ 1 3 4 − λ ε + 2 λ μ 3 ρ θ k 2 1 3 k x − θ t τ τ − ς 0 2 , v 4 ( x , t ) = ± − 3 ρ θ k 2 λ ε + 2 μ λ 1 3 2 − λ ε + 2 λ μ 3 ρ θ k 2 1 3 k x − θ t τ τ − ς 0 .

Case 2: Δ > 0 , β 0 = 0 .

When w > − β 1 , if β 1 > 0 , then the explicit solutions of Eq. (2.7) are

(3.17) w = β 1 2 tanh 2 1 2 β 1 2 ( ζ − ζ 0 ) − β 1 ,

(3.18) w = β 1 2 coth 2 1 2 β 1 2 ( ζ − ζ 0 ) − β 1 ,

then, the corresponding solutions of Eq. (1.1) are

(3.19) u 5 ( x , t ) = 3 θ ε + 2 μ tanh 2 1 2 7 2 1 3 θ 5 3 ρ 1 3 ( k λ ε + 2 μ k λ ) 2 3 − λ ε + 2 λ μ 3 ρ θ k 2 1 3 k x − θ t τ τ − ς 0 − 576 3 θ 5 3 ρ 1 3 ( λ ε k + 2 μ k λ ) 2 3 , v 5 ( x , t ) = ± 6 θ 2 λ ( ε + 2 μ ) tanh 2 1 2 7 2 1 3 θ 5 3 ρ 1 3 ( k λ ε + 2 μ k λ ) 2 3 − λ ε + 2 λ μ 3 ρ θ k 2 1 3 k x − θ t τ τ − ς 0 − 576 3 θ 5 3 ρ 1 3 ( λ ε k + 2 μ k λ ) 2 3 1 2 ,

and

(3.20) u 6 ( x , t ) = 3 θ ε + 2 μ coth 2 1 2 7 2 1 3 θ 5 3 ρ 1 3 ( k λ ε + 2 μ k λ ) 2 3 − λ ε + 2 λ μ 3 ρ θ k 2 1 3 k x − θ t τ τ − ς 0 − 576 3 θ 5 3 ρ 1 3 ( λ ε k + 2 μ k λ ) 2 3 , v 6 ( x , t ) = ± 6 θ 2 λ ( ε + 2 μ ) coth 2 1 2 7 2 1 3 θ 5 3 ρ 1 3 ( k λ ε + 2 μ k λ ) 2 3 − λ ε + 2 λ μ 3 ρ θ k 2 1 3 k x − θ t τ τ − ς 0 − 576 3 θ 5 3 ρ 1 3 ( λ ε k + 2 μ k λ ) 2 3 1 2 .

If β 1 < 0 , then the explicit solution of Eq. (2.7) is

(3.21) w = − β 1 2 tanh 2 1 2 − β 1 2 ( ζ − ζ 0 ) − β 1 ,

then, the corresponding solutions of Eq. (1.1) are

(3.22) u 7 ( x , t ) = 3 θ ε + 2 μ tanh 2 1 2 − 7 2 1 3 θ 5 3 ρ 1 3 ( k λ ε + 2 μ k λ ) 2 3 − λ ε + 2 λ μ 3 ρ θ k 2 1 3 k x − θ t τ τ − ς 0 − 576 3 θ 5 3 ρ 1 3 ( λ ε k + 2 μ k λ ) 2 3 , v 7 ( x , t ) = ± 6 θ 2 λ ( ε + 2 μ ) tanh 2 1 2 − 7 2 1 3 θ 5 3 ρ 1 3 ( k λ ε + 2 μ k λ ) 2 3 − λ ε + 2 λ μ 3 ρ θ k 2 1 3 k x − θ t τ τ − ς 0 − 576 3 θ 5 3 ρ 1 3 ( λ ε k + 2 μ k λ ) 2 3 1 2 .

Case 3: Δ > 0 , β 0 ≠ 0 .

Assume that d < e < c , one of which is 0, and the other two are the roots of F ( w ) , then when d < w < e , the Jacobian elliptic function [64] solution of Eq. (2.7) is

(3.23) w = d + ( e − d ) sn 2 c − d 2 ( ς − ς 0 ) , m .

When w > c , the corresponding solution of Eq. (2.7) is

(3.24) w = − e sn 2 c − d ( ς − ς 0 ) 2 , m + c cn 2 c − d ( ς − ς 0 ) 2 , m ,

where m 2 = e − d c − d .

Under the above conditions, the solutions of Eq. (1.1) can be expressed as

(3.25) u 8 ( x , t ) = − 1 2 λ θ 2 3 3 ρ k 2 ε + 2 μ 1 3 d + ( e − d ) sn 2 c − d 2 − λ ε + 2 λ μ 3 ρ θ k 2 1 3 k x − θ t τ τ − ς 0 , m , v 8 ( x , t ) = ± − 3 ρ θ k 2 λ ε + 2 μ λ 1 3 d + ( e − d ) sn 2 c − d 2 − λ ε + 2 λ μ 3 ρ θ k 2 1 3 k x − θ t τ τ − ς 0 , m ,

and

(3.26) u 9 ( x , t ) = − 1 2 λ θ 2 3 3 ρ k 2 ε + 2 μ 1 3 − e sn 2 c − d 2 − λ ε + 2 λ μ 3 ρ θ k 2 1 3 k x − θ t τ τ − ς 0 , m + c cn 2 c − d 2 − λ ε + 2 λ μ 3 ρ θ k 2 1 3 k x − θ t τ τ − ς 0 , m , v 9 ( x , t ) = ± − 3 ρ θ k 2 λ ε + 2 μ λ 1 3 − e sn 2 c − d 2 − λ ε + 2 λ μ 3 ρ θ k 2 1 3 k x − θ t τ τ − ς 0 , m + c cn 2 c − d 2 − λ ε + 2 λ μ 3 ρ θ k 2 1 3 k x − θ t τ τ − ς 0 , m .

Case 4: Δ < 0 .

When w > 0 , then the solution of Eq. (2.7) is

(3.27) w = 2 β 0 1 + cn β 0 1 4 ( ς − ς 0 ) , m − β 0 ,

where m 2 = 1 2 1 − β 1 2 β 0 .

Similarly, we can obtain the solution of Eq. (1.1) as follows:

(3.28) u 10 ( x , t ) = − 1 2 λ θ 2 3 3 ρ k 2 ε + 2 μ 1 3 12,288 6 ( − b 0 ) 1 2 ρ θ k 2 λ ε + 2 μ λ 1 6 1 + cn 192 12 ( − b 0 ) 1 4 ρ θ k 2 λ ε + 2 μ λ 1 12 − λ ε + 2 λ μ 3 ρ θ k 2 1 3 k x − θ t τ τ − ς 0 , m − 192 6 ( − b 0 ) 1 2 ρ θ k 2 λ ε + 2 μ λ 1 6 , v 10 ( x , t ) = ± − 3 ρ θ k 2 λ ε + 2 μ λ 1 3 12,288 6 ( − b 0 ) 1 2 ρ θ k 2 λ ε + 2 μ λ 1 6 1 + cn 192 12 ( − b 0 ) 1 4 ρ θ k 2 λ ε + 2 μ λ 1 12 − λ ε + 2 λ μ 3 ρ θ k 2 1 3 k x − θ t τ τ − ς 0 , m − 192 6 ( − b 0 ) 1 2 ρ θ k 2 λ ε + 2 μ λ 1 6 .

4 Graphical representation

In this section, the graphical description of the exact solution of DSW Eq. (1.1) is shown in the following figures. The provided solution can help investigate the fluid propagation and understand the practical application of the conformable time fractional DSW equation. By setting the parameter values reasonably and satisfying the conditions (3.5). The soliton solutions in the form of hyperbolic function, trigonometric function and Jacobi Elliptic function were drawn by Maple software. Let λ = 2 , ε = 2 , ρ = 1 , k = 4 , μ = 2 , θ = − 3.031433133 , τ = 0.5 , b 0 = 0.435275282 , then the hyperbolic tangent function solutions of Eq. (1.1) are shown in Figures 1 and 2. Let λ = 2 , ε = 1 , ρ = 2 , k = 1 , μ = 1 , θ = 2.29739671 , τ = 0.5 , b 0 = − 3.031433133 , then the tangent function solution of Eq. (1.1) is shown in Figure 3. Let λ = 3 , ε = 1 , ρ = 2 , k = 1 , μ = 2 , θ = 1.442699906 , τ = 0.5 , b 0 = 0 , then the hyperbolic cotangent solution of Eq. (1.1) is shown in Figure 4. Let λ = 1 , ε = 3 , ρ = 2 , k = 1 , μ = 3 , θ = − 1.782602458 , τ = 0.5 , b 0 = 0 , then the hyperbolic tangent function solutions of Eq. (1.1) are shown in Figures 5 and 6. Let λ = 3 , ε = 2 , ρ = 2 , k = 1 , μ = 1 , θ = 2 , τ = 0.5 , b 0 = − 0.75 , then the Jacobi Elliptic function solution of Eq. (1.1) is shown in Figure 7, and in this figure, the solutions of fractional derivative ( τ = 0.5 ) and integer derivative ( τ = 1 ) were compared. Let λ = 3 , ε = 2 , ρ = 2 , k = 1 , μ = 1 , θ = 2 , τ = 0.5 , and b 0 = − 20.25 , then the Jacobi Elliptic function solution of Eq. (1.1) are shown in Figure 8, which indicates the existence of periodic solutions to the system of Eq. (1.1).

$Figure 1 The soliton solution u 1 ( t , x ) {u}_{1}\left(t,x) of Eq. (1.1) with λ = 2 \lambda =2 , ε = 2 \varepsilon =2 , ρ = 1 \rho =1 , k = 4 k=4 , μ = 2 \mu =2 , θ = ‒ 3.031433133 \theta =‒3.031433133 , τ = 0.5 \tau =0.5 , b 0 = 0.435275282 {b}_{0}=0.435275282 . (a) Three-dimensional graphic, (b) two-dimensional graphic, and (c) contour plot.$

Figure 1

The soliton solution u 1 ( t , x ) of Eq. (1.1) with λ = 2 , ε = 2 , ρ = 1 , k = 4 , μ = 2 , θ = ‒ 3.031433133 , τ = 0.5 , b 0 = 0.435275282 . (a) Three-dimensional graphic, (b) two-dimensional graphic, and (c) contour plot.

$Figure 2 The soliton solution v 1 ( t , x ) {v}_{1}\left(t,x) of Eq. (1.1) with λ = 2 \lambda =2 , ε = 2 \varepsilon =2 , ρ = 1 \rho =1 , k = 4 k=4 , μ = 2 \mu =2 , θ = ‒ 3.031433133 \theta =‒3.031433133 , τ = 0.5 \tau =0.5 , b 0 = 0.435275282 {b}_{0}=0.435275282 . (a) Three-dimensional graphic, (b) two-dimensional graphic, and (c) contour plot.$

Figure 2

The soliton solution v 1 ( t , x ) of Eq. (1.1) with λ = 2 , ε = 2 , ρ = 1 , k = 4 , μ = 2 , θ = ‒ 3.031433133 , τ = 0.5 , b 0 = 0.435275282 . (a) Three-dimensional graphic, (b) two-dimensional graphic, and (c) contour plot.

$Figure 3 The soliton solution v 3 ( t , x ) {v}_{3}\left(t,x) of Eq. (1.1) with λ = 2 \lambda =2 , ε = 1 \varepsilon =1 , ρ = 2 \rho =2 , k = 1 k=1 , μ = 1 \mu =1 , θ = 2.29739671 \theta =2.29739671 , τ = 0.5 \tau =0.5 , b 0 = ‒ 3.031433133 {b}_{0}=‒3.031433133 . (a) Three-dimensional graphic, (b) two-dimensional graphic, and (c) contour plot.$

Figure 3

The soliton solution v 3 ( t , x ) of Eq. (1.1) with λ = 2 , ε = 1 , ρ = 2 , k = 1 , μ = 1 , θ = 2.29739671 , τ = 0.5 , b 0 = ‒ 3.031433133 . (a) Three-dimensional graphic, (b) two-dimensional graphic, and (c) contour plot.

$Figure 4 The soliton solution v 6 ( t , x ) {v}_{6}\left(t,x) of Eq. (1.1) with λ = 3 , ε = 1 , ρ = 2 , k = 1 , μ = 2 , θ = 1.442699906 , τ = 0.5 , b 0 = 0 \lambda =3,\varepsilon =1,\rho =2,k=1,\mu =2,\theta =1.442699906,\tau =0.5,{b}_{0}=0 . (a) Three-dimensional graphic, (b) two-dimensional graphic, and (c) contour plot.$

Figure 4

The soliton solution v 6 ( t , x ) of Eq. (1.1) with λ = 3 , ε = 1 , ρ = 2 , k = 1 , μ = 2 , θ = 1.442699906 , τ = 0.5 , b 0 = 0 . (a) Three-dimensional graphic, (b) two-dimensional graphic, and (c) contour plot.

$Figure 5 The soliton solution u 7 ( t , x ) {u}_{7}\left(t,x) of Eq. (1.1) with λ = 1 , ε = 3 , ρ = 2 , k = 1 , μ = 3 , θ = ‒ 1.782602458 , τ = 0.5 , b 0 = 0 \lambda =1,\varepsilon =3,\rho =2,k=1,\mu =3,\theta =‒1.782602458,\tau =0.5,{b}_{0}=0 . (a) Three-dimensional graphic, (b) two-dimensional graphic, and (c) contour plot.$

Figure 5

The soliton solution u 7 ( t , x ) of Eq. (1.1) with λ = 1 , ε = 3 , ρ = 2 , k = 1 , μ = 3 , θ = ‒ 1.782602458 , τ = 0.5 , b 0 = 0 . (a) Three-dimensional graphic, (b) two-dimensional graphic, and (c) contour plot.

$Figure 6 The soliton solution v 7 ( t , x ) {v}_{7}\left(t,x) of Eq. (1.1) with λ = 1 , ε = 3 , ρ = 2 , k = 1 , μ = 3 , θ = ‒ 1.782602458 , τ = 0.5 , b 0 = 0 \lambda =1,\varepsilon =3,\rho =2,k=1,\mu =3,\theta =‒1.782602458,\tau =0.5,{b}_{0}=0 . (a) Three-dimensional graphic, (b) two-dimensional graphic, and (c) contour plot.$

Figure 6

The soliton solution v 7 ( t , x ) of Eq. (1.1) with λ = 1 , ε = 3 , ρ = 2 , k = 1 , μ = 3 , θ = ‒ 1.782602458 , τ = 0.5 , b 0 = 0 . (a) Three-dimensional graphic, (b) two-dimensional graphic, and (c) contour plot.

$Figure 7 The soliton solution u 8 ( t , x ) {u}_{8}\left(t,x) of Eq. (1.1) with λ = 3 , ε = 2 , ρ = 2 , \lambda =3,\varepsilon =2,\rho =2, k = 1 , μ = 1 , θ = 2 , k=1,\mu =1,\theta =2, b 0 = ‒ 0.75 , {b}_{0}=‒0.75, [ ( a ) ( b ) ( c ) : τ = 0.5 ; ( d ) ( e ) ( f ) : τ = 1 ] \left[\left(a)\left(b)\left(c):\tau =0.5;\left(d)\left(e)(f):\tau =1] . (a) Three-dimensional graphic, (b) two-dimensional graphic, (c) contour plot, (d) three-dimensional graphic, (e) two-dimensional graphic, and (f) contour plot.$

Figure 7

The soliton solution u 8 ( t , x ) of Eq. (1.1) with λ = 3 , ε = 2 , ρ = 2 , k = 1 , μ = 1 , θ = 2 , b 0 = ‒ 0.75 , [ ( a ) ( b ) ( c ) : τ = 0.5 ; ( d ) ( e ) ( f ) : τ = 1 ] . (a) Three-dimensional graphic, (b) two-dimensional graphic, (c) contour plot, (d) three-dimensional graphic, (e) two-dimensional graphic, and (f) contour plot.

$Figure 8 The soliton solution u 10 ( t , x ) {u}_{10}\left(t,x) of Eq. (1.1) with λ = 3 , ε = 2 , ρ = 2 , k = 1 , μ = 1 , θ = 2 , τ = 0.5 , b 0 = − 20.25 \lambda =3,\varepsilon =2,\rho =2,k=1,\mu =1,\theta =2,\tau =0.5,{b}_{0}=-20.25 . (a) Three-dimensional graphic, (b) two-dimensional graphic, and (c) contour plot.$

Figure 8

The soliton solution u 10 ( t , x ) of Eq. (1.1) with λ = 3 , ε = 2 , ρ = 2 , k = 1 , μ = 1 , θ = 2 , τ = 0.5 , b 0 = − 20.25 . (a) Three-dimensional graphic, (b) two-dimensional graphic, and (c) contour plot.

5 Conclusion

This study applies the complete discriminant method of quadratic polynomial to solve the exact soliton solutions of the conformable time-fractional DSW equation. Based on the complete discriminant method, the classification of the roots of quadratic polynomial achieves the classification of the exact soliton solutions of the DSW equation and provides the corresponding expression of the exact solutions. The physical characteristics of the obtained solutions in the shape of spike, multi-wave, kinky lump, interaction lump, M/W-shaped periodic wave have been displayed in 3D, 2D, and Contour maps. Compared with other related research results [16], we provide a new solution method for solving the DSW equation, which is not only fast and effective, but also provides more forms of accurate solutions, making the DSW model better applied in practice, and we also obtained the Jacobian function solution, which has not been reported in previous literature. In future research, in order to better help people understand the DSW system, we will combine the dynamic system analysis method to more comprehensive and in-depth research on the solutions of the DSW equation.

Acknowledgments

The authors are grateful to the anonymous reviewers for their careful reading and useful suggestions, which greatly improved the presentation of the article.

Funding information: The authors state no funding involved.
Author contributions: Da Shi: writing-orginal draft. Zhao Li: writing-review-editing. All authors have accepted responsibility for the entire content of this manuscript and approved its submission.
Conflict of interest: The authors state no conflict of interest.
Data availability statement: Data sharing is not applicable to this article as no datasets were generated or analysed during the current study.

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Received: 2024-06-04

Revised: 2024-07-26

Accepted: 2024-10-18

Published Online: 2024-11-25

This work is licensed under the Creative Commons Attribution 4.0 International License.

Articles in the same Issue

https://doi.org/10.1515/phys-2024-0099

Keywords for this article

Drinfel’d–Sokolov–Wilson equation; fractional order complex transformation; soliton solution; complete discriminant system

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