Soliton dynamics of the KdV–mKdV equation using three distinct exact methods in nonlinear phenomena

M. Atta Ullah; Kashif Rehan; Zahida Perveen; Maasoomah Sadaf; Ghazala Akram

doi:10.1515/nleng-2022-0318

Article Open Access

Soliton dynamics of the KdV–mKdV equation using three distinct exact methods in nonlinear phenomena

M. Atta Ullah , Kashif Rehan , Zahida Perveen , Maasoomah Sadaf and Ghazala Akram

Published/Copyright: April 16, 2024

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From the journal Nonlinear Engineering Volume 13 Issue 1

Abstract

The KdV–mKdV equation is investigated in this study. This equation is a useful tool to model many nonlinear phenomena in the fields of fluid dynamics, quantum mechanics, and soliton wave theory. The exact soliton solutions of the KdV–mKdV equation are extracted using three distinct exact methods, namely, the generalized projective Riccati equation method, the modified auxiliary equation method, and the generalized unified method. Many novel soliton solutions, including kink, periodic, bright, dark, and singular dark–bright soliton solutions, are obtained. Rational functions, exponential functions, trigonometric functions, and hyperbolic functions are contained in the acquired nontrivial exact solutions. The graphical simulation of some obtained solutions is depicted using 3D plots, 2D contour plots, density plots, and 2D line plots. For the first time, the KdV–mKdV equation is investigated using the proposed three exact methods, and many novel solutions, such as dark, bright, and dark–bright singular soliton solutions, are determined, which have never been reported in the literature.

Keywords: generalized Riccati equation method; modified auxiliary equation method; generalized unified method; exact solutions

1 Introduction

Every natural phenomenon involves many nonlinearities and varying factors. Nonlinear partial differential equations (NLPDEs) are significant mathematical tools for the complete representation of several natural phenomena and varying mechanisms. NLPDEs are utilized by various engineering fields and applied sciences, including plasma physics, mathematical biology, biophysics, optical fibers, solid-state physics, fluid dynamics, nonlinear optics, and population dynamics. Many NLPDEs, such as the Boiti–Leon–Manna–Pempinelli equation [1], the time-fractional nonlinear diffusion wave equation [2], the M-truncated paraxial wave equation [3], and the time-variable Kadomtsev–Petviashvili equation [4], provide the best tools to represent different nonlinear phenomena and dynamical processes. A variety of physical phenomena and dynamical processes that exhibit nonlinear behavior are modeled using partial differential such NLPDEs.

To understand different nonlinear physical phenomena and dynamical processes, it is necessary to extract the exact solutions of NLPDEs. Due to the importance of understanding nonlinear physical phenomena and dynamical processes, the construction of exact solutions to NPDEs has been a major focus of research. Since it is impossible to solve all NLPDEs using a unified approach, researchers have developed various exact methods to obtain the exact solutions of NLPDEs. In recent decades, many exact methods have been developed and practiced by various researchers to obtain exact solutions of NLPDEs. These methods include extended ( G ′ G 2 ) -expansion method [5], modified extended direct algebraic method [6], generalized projective Riccati equation (GPRE) method [7], modified auxiliary equation (MAE) method [8,9], generalized unified (GU) method [10], hyperbolic and exponential ansatz methods [11], multiple Exp-function method [12], Tikhonov regularization method [13], and Hirota bilinear method [14]. These methods have enabled researchers to obtain exact solutions of NLPDEs.

In this study, GPRE method, MAE method, and GU method are utilized to extract the exact soliton solutions of the KdV–mKdV equation. These methods are effective and have been utilized by many researchers to investigate NLPDEs. The GPRE method [15,16] has been used by various researchers to extract the exact soliton solutions of several NLPDEs. The MAE method [17–19] has also been extensively used by researchers to extract the exact soliton solutions of different NLPDEs. In the study by Akram et al. [17,18], MAE method is explained in detail and applied to an Lakshmanan-Porsezian-Daniel model along with three nonlinearity laws. Using the MAE method, different soliton solutions are successfully extracted. In the study by Akram et al. [19], the KdV equation and KdV–Burger equation are solved using the MAE method, and many soliton solutions are obtained. The GU method [20–22] has been utilized by various researchers to obtain one- and multi-soliton solutions of NLPDEs. In the study by Osman [20], the GU method is extensively used to acquire single-soliton and multi-soliton solutions of the KdV equation and the ( 2 + 1 ) -dimensional Nizhnik–Novikov–Veselov system. In the study by Osman [21], ( 2 + 1 ) -dimensional Bogoyavlensky–Konopelchenko equation is investigated using the GU method, and different types of soliton rational solutions are obtained, which exhibit kink, anti-kink, and periodic-kink soliton behavior. In the study by Osman [22], KdV–Sawada–Kotera–Ramani equation is investigated using the GU method, and many single-soliton solutions and multi-soliton solutions are retrieved.

In this article, KdV–mKdV equation is considered to be investigated for the extraction of exact soliton solutions. One of the key applications of the KdV–mKdV equations is in the study of solitons. The KdV–mKdV equation has been used to model solitons in a variety of physical systems including water waves, plasma waves, and nonlinear optics. The KdV–mKdV equations have been used to study the emergence of coherent structures in turbulent flows, and these are shown to be useful to describe the dynamics of these structures. Therefore, KdV–mKdV is an important structure to be examined for exact solutions. The mathematical form of the proposed KdV–mKdV equation is stated, as

(1) Δ t + ( h 3 Δ 2 + h 2 Δ + h 1 ) Δ x + u x x x = 0 .

In this study, the KdV–mKdV equation is considered to be investigated for the extraction of exact soliton solutions. The wave profile depicted by Δ in Eq. (1) is a function of the space variable x and temporal variable t . This equation is derived from the generalized KdV–mKdV equation [23]. The KdV and mKdV equations are the most popular soliton equations, and several studies have been conducted on them. Nonlinear terms of KdV and mKdV equations are often present together in practical problems such as fluid physics and quantum field theory, and the combined KdV–mKdV equation describes the wave propagation of bound particles, sound waves, and thermal pulses [24,25]. Various exact methods such as the inverse scattering method, homogeneous balance method, Hirota bilinear method, and unified algebraic method have been used to extract exact soliton solutions of the KdV–mKdV equation.

The limitation of this work corresponds to the limitation of the considered model and three exact methods. The KdV–mKdV equation is a useful model for certain types of shallow water waves, but its applicability and accuracy are not suitable for all types of wave patterns. This equation is a simple model and does not take into account all complex dynamics. The GPRE method has several advantages but also has some limitations. Although it is effective for some specific NLPDEs, it is not suitable for other types of complex NLPDEs. Similarly, the MAE and GU methods are very effective and useful exact techniques, but both have some limitations. Both methods may not be applicable to each type of NLPDE. For some specific NLPDEs, these methods are significant in extracting exact solutions.

The aim of this article is to investigate the KdV–mKdV equation for its exact solutions. To extract exact soliton solutions, three exact methods, namely, GPRE method, MAE method, and GU method, are utilized. These methods are simple, reliable, and effective and are used to obtain the exact solution of the proposed equation. The KdV–mKdV equation is investigated by these exact methods for the first time, and many novel exact soliton solutions of the KdV–mKdV equation are obtained using the proposed exact methods, which have never been found in the literature.

For better understanding and identification of the pattern of the wave, retrieved solutions are plotted at the end of this article for some specific values of parameters. The graphical simulations of acquired solutions represent different soliton wave patterns. Based on these patterns of soliton waves, obtained solutions are categorized into singular soliton solutions, dark soliton solutions, bright soliton solutions, dark–bright singular soliton solutions, kink soliton solutions, and many others.

The sections of the research article are as follows: Section 2 presents the demarcation of the GPRE method, MAE method, and GU method. In Section 3, the proposed methods are applied. The graphical interpretation of the obtained solutions is presented in Section 4. The discussion and comparison of the obtained results are presented in Section 5. The results of this study are summarized in Section 6 of this article.

2 Description of methods

For an unknown function Δ ( x , t ) , NPDE is considered, as

(2) G ( Δ , Δ x , Δ t , Δ x x , Δ t t , Δ x t , … ) = 0 ,

where Δ depicts the wave profile function of x and t . The NPDE (2) is transformed into ordinary differential equation (ODE) of the form:

(3) H ( Γ , Γ ′ , Γ ″ , … ) = 0 ,

using wave transformation Γ = Γ ( Λ ) , where Λ = μ x − θ t + ϑ .

2.1 GPRE method

The formal solution to Eq. (3) that is illustrated in previous studies [15,16] is given as follows:

(4) Γ ( Λ ) = a 0 + ∑ i = 1 M φ i − 1 ( a i φ ( Λ ) + b i Y ( Λ ) ) ,

whre the unknown constants a 0 , a i , and b i are to be evaluated later. The auxiliary equations for the function of φ and Y are expressed, as

(5) d φ d Λ = − τ φ Y , d Y d Λ = − τ Y 2 − p φ + G ,

where

(6) Y 2 = − τ G − 2 p φ + ( p 2 + j ) φ 2 G , τ = ± ( 1 ) , j = ± ( 1 ) ,

where G and p are the real-valued constants. The homogeneous balance principle (HBP) is used to extract the value of M , as clearly explained in the study by Akram et al. [5]. Eq. (4) will be substituted into the transformed ODE (3). The values of the unknowns can be extracted by solving the system of linear equations obtained by setting the coefficients of φ m Y n equal to zero, where ( m , n = 0 , 1 , 2 , 3 , … ) . The auxiliary equations from Eq. (5) have the following solutions:

Case 1. If G > 0 , τ = − 1 , and j = − 1 ,

(7) φ 1 ( Λ ) = G sech ( G Λ ) p sech ( G Λ ) + 1 , Y 1 ( Λ ) = G tanh ( G Λ ) p sech ( G Λ ) + 1 .

If G > 0 , τ = − 1 , and j = 1 ,

(8) φ 2 ( Λ ) = G csch ( G Λ ) p csc ( G Λ ) + 1 , Y 2 ( Λ ) = G coth ( G Λ ) p csch ( G Λ ) + 1 .

Case 2. If G > 0 , τ = 1 , and j = − 1 ,

(9) φ 3 ( Λ ) = G sec ( G Λ ) p sec ( G Λ ) + 1 , Y 3 ( Λ ) = G tan ( G Λ ) p sec ( G Λ ) + 1 .

If G > 0 , τ = 1 , and j = 1 ,

(10) φ 4 ( Λ ) = G csc ( G Λ ) p csc ( G Λ ) + 1 , Y 4 ( Λ ) = − G cot ( G Λ ) p csc ( G Λ ) + 1 .

Case 3. If G = 0 and p = 0 ,

(11) φ 5 ( Λ ) = D Λ , Y 5 ( Λ ) = 1 τ Λ .

The exact soliton solutions of Eq. (1) are obtained by substituting values of unknown parameters in (4) with Eqs. (7)–(11).

2.2 MAE method

In the MAE method, the general solution of Eq. (1) is expressed, as

(12) Γ ( Λ ) = L 0 + ∑ i = 1 M ( L i K i w ( Λ ) + Q i K − i w ( Λ ) ) ,

where L 0 , L i , and Q i are the constants to be evaluated. The function W ( Λ ) is defined as

(13) d d Λ w ( Λ ) = h 7 + h 8 K w ( Λ ) + h 9 K − w ( Λ ) ln ( K ) ,

where h 7 , h 8 , and h 9 are the constants parameters to be evaluated. The value of M in Eq. (12) is to be calculated by HBP, which is clearly expressed in the study by Akram et al. [5]. The obtained formal solution of Eq. (3) with its auxiliary Eq. (13) will be inserted in transformed ODE (3). The values of unknown parameters will be obtained by solving the system of equations, which are obtained by collecting all coefficients of K i w for ( i = 0 , ± 1 , ± 2 , … ) and setting them equal to zero. The solution of auxiliary Eq. (13) is given as follows:

Case 1. If − h 7 2 + 4 h 8 h 9 < 0 and h 8 ≠ 0 ,

(14) K w = − h 7 + − h 7 2 + 4 h 8 h 9 tan 1 2 − h 7 2 + 4 h 8 h 9 Λ 2 h 8

(15) K w = − h 7 + − h 7 2 + 4 h 8 h 9 cot 1 2 − h 7 2 + 4 h 8 h 9 Λ 2 h 8 .

Case 2. If − h 7 2 + 4 h 8 h 9 > 0 and h 8 ≠ 0 ,

(16) K w = − h 7 + − h 7 2 + 4 h 8 h 9 tanh 1 2 − h 7 2 + 4 h 8 h 9 Λ 2 h 8

(17) K w = − h 7 + − h 7 2 + 4 h 8 h 9 coth 1 2 − h 7 2 + 4 h 8 h 9 Λ 2 h 8 .

Case 3. If − h 7 2 + 4 h 8 h 9 = 0 and h 8 ≠ 0 ,

(18) K w = − h 7 Λ + 2 2 h 8 Λ .

The exact soliton solutions of Eq. (1) are obtained by substituting values of unknown parameters in Eq. (12) with Eqs. (14)–(18).

2.3 GU method

Single soliton formal solution in the GU method which has been clearly mentioned in [21,22] is defined as

(19) Γ ( Λ ) = l 1 + l 2 χ ( Λ ) l 3 + l 4 χ ( Λ ) ,

where l 1 , l 2 , l 3 , and l 4 are unknown constant parameters which are to be evaluated later. The function χ ( Λ ) is defined as

(20) χ = e c 1 Λ ,

where c 1 is the unknown constant to be evaluated. The unknown constant c 1 is evaluated by inserting the formal solution of Eq. (19) with its auxiliary Eq. (20) into the transformed ODE (3). The values of the unknown parameters are obtained by solving the system of equations that arise from collecting all coefficients of χ i ( Λ ) for ( i = 0 , ± 1 , ± 2 , … ) and setting them equal to zero. The obtained values of the unknown parameters are then substituted into the formal solution Eq. (19) with auxiliary Eq. (20). As a result, the solution to the considered NLPDE is obtained. In the following section, the proposed exact solutions are applied.

3 Application of proposed methods

The ODE of the following form is obtained using the wave transformation Λ = μ x − θ t + ϑ , which is given as

(21) θ Γ ′ + ( h 3 Γ 2 + h 2 Γ + h 1 ) μ Γ ′ + μ 3 Γ ‴ = 0 ,

where Γ is the unknown wave function d Γ d Λ .

3.1 Application of the GPRE method

In Eq. (4), the value of M is 1, which is extracted by HBP. The formal solution of Eq. (21) is written as

(22) Γ ( Λ ) = a 0 + a 1 φ ( Λ ) + b 1 Y ( Λ ) ,

where a 0 , a 1 , and b 1 are the constant parameters. Eq. (22) with Eq. (5) and Eq. (6) is inserted in Eq. (21). The coefficients of φ m Y n for ( m , n = 0 , 1 , 2 , 3 , … ) are accumulated. The system of homogeneous equations is acquired by putting all coefficients equal to zero. The following values of unknown parameters are obtained by solving the systems of homogeneous equations.

Set 1.

θ = 1 24 p 2 τ h 3 + 12 j τ h 3 ( − 32 B μ 3 p 2 τ 4 h 3 + 12 G μ 3 p 2 τ 4 h 3 + 24 G j μ 3 τ 4 h 3 − 16 B μ 3 p 2 τ 2 h 3 + 12 G μ 3 p 2 τ 2 h 3 + 12 G j μ 3 τ 2 h 3 − 2 B μ 3 p 2 h 3 − 12 μ p 2 τ h 1 h 3 + 3 μ p 2 τ h 2 2 − 12 j μ τ h 1 h 3 + 3 j μ τ h 2 2 ) , a 1 = ± − − 6 p 2 τ − 6 j τ B h 3 τ μ , a 0 = ± 1 2 − − 6 p 2 τ − 6 j τ B h 3 h 3 × 8 μ p τ 2 + − − 6 p 2 τ − 6 j τ B h 3 h 2 + 2 μ p , μ = μ , b 1 = 0 .

The following family of solutions are acquired by taking values of parameters from Set 1:

Family 1. The formal solution 22 for Set 1 is written, as

(23) Δ ( x , t ) = ± 1 2 − − 6 p 2 τ − 6 j τ B h 3 h 3 ( 8 μ p τ 2 + − − 6 p 2 τ − 6 j τ B h 3 h 2 + 2 μ p ± − − 6 p 2 τ − 6 j τ B h 3 τ μ φ ( Λ ) .

Case 1. If G > 0 , τ = − 1 , and j = − 1 ,

(24) Δ 1 1 ( x , t ) = ± 1 2 − − 6 p 2 τ − 6 j τ B h 3 h 3 × 8 μ p τ 2 + − − 6 p 2 τ − 6 j τ B h 3 h 2 + 2 μ p ± − − 6 p 2 τ − 6 j τ B h 3 τ μ × G sech ( G ( μ x − θ t + ϑ ) ) p sech ( G ( μ x − θ t + ϑ ) ) + 1 .

If G > 0 , τ = − 1 , and j = 1 ,

(25) Δ 2 1 ( x , t ) = ± 1 2 − − 6 p 2 τ − 6 j τ B h 3 h 3 × 8 μ p τ 2 + − − 6 p 2 τ − 6 j τ B h 3 h 2 + 2 μ p ± − − 6 p 2 τ − 6 j τ B h 3 τ μ × G csch ( G ( μ x − θ t + ϑ ) ) p csch ( G ( μ x − θ t + ϑ ) ) + 1 .

Case 2. If G > 0 , τ = 1 , and j = − 1 ,

(26) Δ 3 1 ( x , t ) = ± 1 2 − − 6 p 2 τ − 6 j τ B h 3 h 3 × 8 μ p τ 2 + − − 6 p 2 τ − 6 j τ B h 3 h 2 + 2 μ p ± − − 6 p 2 τ − 6 j τ B h 3 τ μ × G sec ( G ( μ x − θ t + ϑ ) ) p sec ( G ( μ x − θ t + ϑ ) ) + 1 .

If G > 0 , τ = 1 , and j = 1 ,

(27) Δ 4 1 ( x , t ) = ± 1 2 − − 6 p 2 τ − 6 j τ B h 3 h 3 × 8 μ p τ 2 + − − 6 p 2 τ − 6 j τ B h 3 h 2 + 2 μ p ± − − 6 p 2 τ − 6 j τ B h 3 τ μ × G csc ( G ( μ x − θ t + ϑ ) ) p csc ( G ( μ x − θ t + ϑ ) ) + 1 .

Case 3. If G = 0 and p = 0 ,

(28) Δ 5 1 ( x , t ) = ± 1 2 − − 6 p 2 τ − 6 j τ B h 3 h 3 × 8 μ p τ 2 + − − 6 p 2 τ − 6 j τ B h 3 h 2 + 2 μ p ± − − 6 p 2 τ − 6 j τ B h 3 τ μ D μ x − θ t + ϑ ,

where D is the real-valued constant. In the following subsection, the MAE method is applied to extract the solution of proposed equation.

3.2 Application of the MAE method

In Eq. (12), the value of M is 1, which is extracted by HBP. The formal solution of Eq. (21) is written, as

(29) Γ ( Λ ) = L 0 + L 1 K W ( Λ ) + Q 1 K − W ( Λ ) ,

where L 0 , L 1 , and Q 1 are the constant parameters. Eq. (29) with Eq. (13) is inserted in Eq. (21). The coefficients of K i W for ( i = 0 , 1 , 2 , 3 , … ) are accumulated. The system of homogeneous equations is acquired by putting all coefficients equal to zero. The following values of unknown parameters are obtained by solving the systems of homogeneous equations.

Set 1.

μ = μ , θ = 1 24 μ 60 μ 2 h 7 2 − 48 μ 2 h 8 h 9 − 24 h 1 + 6 h 2 2 h 3 , ϑ = ϑ , L 0 = ± 3 − 6 h 3 h 3 μ h 7 + 1 6 h 2 − 6 h 3 , L 1 = ± − 6 h 3 h 9 μ , Q 1 = 0 .

Set 2.

μ = μ , θ = 1 24 μ 60 μ 2 h 7 2 − 48 μ 2 h 8 h 9 − 24 h 1 + 6 h 2 2 h 3 , ϑ = ϑ , L 0 = ± 3 − 6 h 3 h 3 μ h 7 + 1 6 h 2 − 6 h 3 , L 1 = 0 , Q 1 = ± − 6 h 3 h 9 μ .

The following family of solutions are acquired by taking values of parameters from Set 1:

Family 1. The formal solution from Eq. (29) for Set 1 is written, as

Δ ( x , t ) = ± 3 − 6 h 3 h 3 μ h 7 + 1 6 h 2 − 6 h 3 ± − 6 h 3 h 9 μ K W ( Λ ) .

Case 1. If − h 7 2 + 4 h 8 h 9 < 0 and h 8 ≠ 0 ,

(30) Δ 1 2 ( x , t ) = ± 3 − 6 h 3 h 3 μ h 7 + 1 6 h 2 − 6 h 3 ± − 6 h 3 h 9 μ × − h 7 + − h 7 2 + 4 h 8 h 9 tan 1 2 − h 7 2 + 4 h 8 h 9 ( μ x − θ t + ϑ ) 2 h 8

(31) Δ 2 2 ( x , t ) = ± 3 − 6 h 3 h 3 μ h 7 + 1 6 h 2 − 6 h 3 ± − 6 h 3 h 9 μ × − h 7 − − h 7 2 + 4 h 8 h 9 cot 1 2 − h 7 2 + 4 h 8 h 9 ( μ x − θ t + ϑ ) 2 h 8 .

Case 2. If − h 7 2 + 4 h 8 h 9 > 0 and h 8 ≠ 0 ,

(32) Δ 3 2 ( x , t ) = ± 3 − 6 h 3 h 3 μ h 7 + 1 6 h 2 − 6 h 3 ± − 6 h 3 h 9 μ × − h 7 − − h 7 2 + 4 h 8 h 9 tanh 1 2 − h 7 2 + 4 h 8 h 9 ( μ x − θ t + ϑ ) 2 h 8

(33) Δ 4 2 ( x , t ) = ± 3 − 6 h 3 h 3 μ h 7 + 1 6 h 2 − 6 h 3 ± − 6 h 3 h 9 μ × − h 7 − − h 7 2 + 4 h 8 h 9 coth 1 2 − h 7 2 + 4 h 8 h 9 ( μ x − θ t + ϑ ) 2 h 8 .

Case 3. If − h 7 2 + 4 h 8 h 9 = 0 and h 8 ≠ 0 ,

(34) Δ 5 2 ( x , t ) = ± 3 − 6 h 3 h 3 μ h 7 + 1 6 h 2 − 6 h 3 ± − 6 h 3 h 9 μ − h 7 ( μ x − θ t + ϑ ) + 2 2 h 8 ( μ x − θ t + ϑ ) .

The following family of solutions are acquired by taking values of parameters from Set 2:

Family 2. The formal solution from Eq. (29) for Set 1 is written, as

Δ ( x , t ) = ± 3 − 6 h 3 h 3 μ h 7 + 1 6 h 2 − 6 h 3 ± − 6 h 3 h 9 μ K − W ( Λ ) .

Case 1. If − h 7 2 + 4 h 8 h 9 < 0 and h 8 ≠ 0 ,

(35) Δ 1 3 ( x , t ) = ± 3 − 6 h 3 h 3 μ h 7 + 1 6 h 2 − 6 h 3 ± − 6 h 3 h 9 μ × − h 7 + − h 7 2 + 4 h 8 h 9 tan 1 2 − h 7 2 + 4 h 8 h 9 ( μ x − θ t + ϑ ) 2 h 8 − 1

(36) Δ 2 3 ( x , t ) = ± 3 − 6 h 3 h 3 μ h 7 + 1 6 h 2 − 6 h 3 ± − 6 h 3 h 9 μ × − h 7 − − h 7 2 + 4 h 8 h 9 cot 1 2 − h 7 2 + 4 h 8 h 9 ( μ x − θ t + ϑ ) 2 h 8 − 1 .

Case 2. If − h 7 2 + 4 h 8 h 9 > 0 and h 8 ≠ 0 ,

(37) Δ 3 3 ( x , t ) = ± 3 − 6 h 3 h 3 μ h 7 + 1 6 h 2 − 6 h 3 ± − 6 h 3 h 9 μ × − h 7 − − h 7 2 + 4 h 8 h 9 tanh 1 2 − h 7 2 + 4 h 8 h 9 ( μ x − θ t + ϑ ) 2 h 8 − 1

(38) Δ 4 3 ( x , t ) = ± 3 − 6 h 3 h 3 μ h 7 + 1 6 h 2 − 6 h 3 ± − 6 h 3 h 9 μ × − h 7 − − h 7 2 + 4 h 8 h 9 coth 1 2 − h 7 2 + 4 h 8 h 9 ( μ x − θ t + ϑ ) 2 h 8 − 1 .

Case 3. If − h 7 2 + 4 h 8 h 9 = 0 and h 8 ≠ 0 ,

(39) Δ 5 3 ( x , t ) = ± 3 − 6 h 3 h 3 μ h 7 + 1 6 h 2 − 6 h 3 ± − 6 h 3 h 9 μ − h 7 ( μ x − θ t + ϑ ) + 2 2 h 8 ( μ x − θ t + ϑ ) − 1 .

In the following subsection, the MAE method is applied to extract the solution of proposed equation.

3.3 Application of the GU method

In the application of the GU method, Eq. (19) with Eq. (20) is inserted in Eq. (21). The coefficients of χ i ( Λ ) for ( i = 0 , 1 , 2 , 3 , … ) are accumulated. The system of homogeneous equations is acquired by putting all coefficients equal to zero. The following values of unknown parameters are obtained by solving the systems of homogeneous equations.

Set 1.

c 1 = ± 1 μ − − 12 μ h 1 h 3 + 3 μ h 2 2 − 12 θ h 3 14 μ h 3 , l 3 = l 3 , l 4 = l 4 , l 1 = ± ( 4 θ h 3 + 4 μ h 1 h 3 + μ h 2 2 ) 7 l 3 14 h 3 − μ ( 4 μ h 1 h 3 − μ h 2 2 + 4 θ h 3 ) + l 3 h 2 2 h 3 , l 2 = ± ( ± 7 μ h 2 + 5 − 28 μ 2 h 1 h 3 + 7 μ 2 h 2 2 − 28 μ θ h 3 ) l 4 14 μ h 3 .

The following family of solutions are acquired by taking values of parameters from Set 1:

Family 1. The formal solution from Eq. (19) for Set 1 is written, as

(40) Δ 1 4 ( x , t ) = 1 l 3 + l 4 e c 1 Λ ± ( 4 θ h 3 + 4 μ h 1 h 3 + μ h 2 2 ) 7 l 3 14 h 3 − μ ( 4 μ h 1 h 3 − μ h 2 2 + 4 θ h 3 ) + l 3 h 2 2 h 3 ± ( ± 7 μ h 2 + 5 − 28 μ 2 h 1 h 3 + 7 μ 2 h 2 2 − 28 μ θ h 3 ) l 4 14 μ h 3 e c 1 Λ .

4 Graphical illustration of solutions

In this section, the soliton solutions extracted from KdV–mKdV are explained using plotted graphs. The line graphs, contour graphs, density graphs, and surface 3D-graphs of the obtained soliton solutions are included. The soliton solutions obtained include trigonometric functions, hyperbolic functions, exponential functions, and rational functions. The kink solitons, periodic soliton, dark soliton, bright soliton, and singular dark–bright soliton solutions are depicted in the graphs of the obtained solutions. These graphs are generated by assigning suitable values to the parameters included in the obtained solutions. The values of known parameters can be taken from the corresponding set of the family to plot the graphs of obtained solutions. The vertical axis in the plotted 3D graphs shows the values of function Δ ( x , t ) . For the line graphs of the solutions, the value of the temporal variable t = 1 is taken. The plotted graphs of the obtained soliton solutions are represented in Figures 1, 2, 3, 4, 5.

$Figure 1 Graphical representation of soliton solution Δ 1 1 ( x , t ) {\Delta }_{1}^{1}\left(x,t) : h 3 = ‒ 3 {h}_{3}=‒3 , h 8 = h 9 = 0.01 {h}_{8}={h}_{9}=0.01 , τ = j = ‒ 1 \tau =j=‒1 , p = 2 p=2 , μ = 0.3 \mu =0.3 , rest of the constant parameters = 1, positive value of a 1 {a}_{1} is taken. 3D surface, 2D contour, density, and line graphs are shown in (a), (b), (c), and (d), respectively.$

Figure 1

Graphical representation of soliton solution Δ 1 1 ( x , t ) : h 3 = ‒ 3 , h 8 = h 9 = 0.01 , τ = j = ‒ 1 , p = 2 , μ = 0.3 , rest of the constant parameters = 1, positive value of a 1 is taken. 3D surface, 2D contour, density, and line graphs are shown in (a), (b), (c), and (d), respectively.

$Figure 2 Graphical representation of soliton solution Δ 1 1 ( x , t ) {\Delta }_{1}^{1}\left(x,t) : h 3 = ‒ 3 {h}_{3}=‒3 , h 8 = h 9 = 0.01 {h}_{8}={h}_{9}=0.01 , τ = j = ‒ 1 \tau =j=‒1 , p = 2 p=2 , μ = 0.3 \mu =0.3 , rest of the constant parameters = 1, negative value of a 1 {a}_{1} is taken. 3D surface, 2D contour, density, and line graphs are shown in (a), (b), (c), and (d), respectively.$

Figure 2

Graphical representation of soliton solution Δ 1 1 ( x , t ) : h 3 = ‒ 3 , h 8 = h 9 = 0.01 , τ = j = ‒ 1 , p = 2 , μ = 0.3 , rest of the constant parameters = 1, negative value of a 1 is taken. 3D surface, 2D contour, density, and line graphs are shown in (a), (b), (c), and (d), respectively.

$Figure 3 Graphical representation of soliton solution Δ 4 1 ( x , t ) {\Delta }_{4}^{1}\left(x,t) : h 3 = ‒ 3 {h}_{3}=‒3 , h 8 = h 9 = 0.01 {h}_{8}={h}_{9}=0.01 , τ = j = ‒ 1 \tau =j=‒1 , p = 2 p=2 , μ = 0.3 \mu =0.3 , rest of the constant parameters = 1. 3D surface, 2D contour, density, and line graphs are shown in (a), (b), (c), and (d), respectively.$

Figure 3

Graphical representation of soliton solution Δ 4 1 ( x , t ) : h 3 = ‒ 3 , h 8 = h 9 = 0.01 , τ = j = ‒ 1 , p = 2 , μ = 0.3 , rest of the constant parameters = 1. 3D surface, 2D contour, density, and line graphs are shown in (a), (b), (c), and (d), respectively.

$Figure 4 Graphical representation of soliton solution Δ 4 1 ( x , t ) {\Delta }_{4}^{1}\left(x,t) : h 1 = h 3 = ‒ 1 , {h}_{1}={h}_{3}=‒1, rest of the constant parameters = 1. 3D surface, 2D contour, density, and line graphs are shown in (a), (b), (c), and (d), respectively.$

Figure 4

Graphical representation of soliton solution Δ 4 1 ( x , t ) : h 1 = h 3 = ‒ 1 , rest of the constant parameters = 1. 3D surface, 2D contour, density, and line graphs are shown in (a), (b), (c), and (d), respectively.

$Figure 5 Graphical representation of soliton solution Δ 3 3 ( x , t ) {\Delta }_{3}^{3}\left(x,t) : h 3 = ‒ 1 , {h}_{3}=‒1, rest of the constant parameters = 1. 3D surface, 2D contour, density, and line graphs are shown in (a), (b), (c), and (d), respectively.$

Figure 5

Graphical representation of soliton solution Δ 3 3 ( x , t ) : h 3 = ‒ 1 , rest of the constant parameters = 1. 3D surface, 2D contour, density, and line graphs are shown in (a), (b), (c), and (d), respectively.

5 Results and discussion

The KdV–mKdV model was investigated in this study using three exact methods, namely, the GPRE method, MAE method, and GU method. The exact soliton solutions of the proposed equation were obtained using these methods, which included various functions such as trigonometric, hyperbolic, exponential, and rational functions. Many wave patterns were extracted from these exact soliton solutions, with the majority of solutions obtained from the MAE and GPRE methods.

Constructing various soliton solutions using these mathematical techniques is possible. Exact soliton solutions of the KdV–mKdV equation are useful in understanding wave structures in physical systems governed by the equation. Graphical representation is a useful tool for illustrating these wave structures, and to this end, surface 3D-graphs, density graphs, contour graphs, and line graphs are presented in Section 4. Suitable parameter choices were used to plot these graphs, which depict various phenomena involving the KdV–mKdV equation, including kink, periodic, bright, dark, and singular dark–bright solitons.

The bright and dark solitons for negative and positive values of ± a 1 in the exact soliton solution Δ 1 1 are depicted in Figures 1 and 2, respectively. Figure 3 shows the periodic soliton by exact soliton solution Δ 4 1 , while Figure 4 depicts the kink soliton by exact soliton solution Δ 1 4 . Finally, Figure 5 shows the dark–bright soliton by exact soliton solution Δ 3 3 .

6 Conclusion

In this article, the KdV–mKdV equation is investigated using three exact techniques, namely, the GPRE method, MAE method, and GU method. Many novel exact soliton solutions of the proposed model are obtained. The description of these methods is presented in detail in this article, aiding in the understanding of the solutions of the equation. Only nontrivial solutions are considered, and the graphical interpretation of the obtained solutions is demonstrated through line graphs, contour graphs, density graphs, and surface 3D-graphs. The types of soliton solutions such as kink, periodic, bright, dark, singular dark–bright, and many other singular soliton solutions can be determined through graphical simulations of the obtained solutions, which contain trigonometric, hyperbolic, exponential, and rational functions. These functions exhibit special properties that are necessary for solitons such as being periodic, having exponential decay, and a balance between nonlinearity and dispersion. Kink soliton solutions are characterized by a sharp transition between two different stable states, while periodic soliton solutions have a repeating structure. Bright and dark soliton solutions are distinguished by their intensity profile, with bright solitons having a peak in intensity and dark solitons having a dip. Finally, a dark–bright soliton is a type of soliton that exhibits both dark and bright features in its structure. It consists of a localized region of decreased intensity or amplitude (dark soliton) embedded within a localized region of increased intensity or amplitude (bright soliton).

The novelty of the results presented in this study highlights its significance. The KdV–mKdV equation is investigated using these three methods for the first time in the literature, and many novel soliton solutions, such as dark, bright, and dark–bright soliton solutions, are obtained using the proposed methods. Further investigations into the KdV–mKdV equation using alternative methods may be conducted in the future. Therefore, there is still a significant amount of work that needs to be done on this model. Additional research may provide new insights and improve our understanding of the behavior and properties of the KdV–mKdV equation. This could lead to the development of more accurate and efficient mathematical models and numerical methods for solving this equation, which could have important applications in various fields, including fluid dynamics, plasma physics, and nonlinear optics.

Funding information: The authors state no funding involved.
Author contributions: All authors have accepted responsibility for the entire content of this manuscript and approved its submission.
Conflict of interest: The authors declare that they have no known competing financial interest or personal relationships that could have appeared to influence the work reported in this article.
Data availability statement: No data associated with this manuscript.

References

[1] Ismael HF, Akkilic AN, Murad MAS, Bulut H, Mahmoud W, Osman MS. Boiti-Leon-Manna-Pempinelli equation including time-dependent coefficient (vcBLMPE): a variety of nonautonomous geometrical structures of wave solutions. Nonlinear Dyn. 2022;110:3699–712. 10.1007/s11071-022-07817-5Search in Google Scholar

[2] Arqub OA, Tayebi S, Baleanu D, Osman MS, Mahmoud W, Alsulami H. A numerical combined algorithm in cubic B-spline method and finite difference technique for the time-fractional nonlinear diffusion wave equation with reaction and damping terms. Results Phys. 2022;41:105912. 10.1016/j.rinp.2022.105912Search in Google Scholar

[3] Rasool T, Hussain R, Al SMA, Mahmoud W, Osman MS. A variety of optical soliton solutions for the M-truncated paraxial wave equation using Sardar-subequation technique. Optical Quantum Electronics. 2023;55(5):396. 10.1007/s11082-023-04655-6Search in Google Scholar

[4] Ismael HF, Sulaiman TA, Nabi HR, Mahmoud W, Osman MS. Geometrical patterns of time variable Kadomtsev–Petviashvili (i) equation that models dynamics of waves in thin films with high surface tension. Nonlinear Dyn. 2023;111:9457–66. 10.1007/s11071-023-08319-8Search in Google Scholar

[5] Akram G, Sadaf M, Khan MAU. Soliton dynamics of the generalized shallow water like equation in nonlinear phenomenon. Frontiers Phys. 2022;10:140. 10.3389/fphy.2022.822042Search in Google Scholar

[6] Akram G, Sadaf M, Khan MAU. Dynamics investigation of the (4+1) -dimensional Fokas equation using two effective techniques. Results Phys. 2022;42:105994. 10.1016/j.rinp.2022.105994Search in Google Scholar

[7] Zayed EME, Alurrfi KAE. The generalized projective Riccati equations method for solving nonlinear evolution equations in mathematical physics. Abstr Appl Anal. 2014;2014:259190. 10.1155/2014/259190Search in Google Scholar

[8] Khater MMA, Lu D, Attia RAM. Dispersive long wave of nonlinear fractional Wu-Zhang system via a modified auxiliary equation method. AIP Advances. 2019;9(2):025003. 10.1063/1.5087647Search in Google Scholar

[9] Khater MMA, Attia RAM, Lu D. Modified auxiliary equation method versus three nonlinear fractional biological models in present explicit wave solutions. Math Comput Appl. 2018;24(1):1. 10.3390/mca24010001Search in Google Scholar

[10] Osman MS, Machado JAT. New nonautonomous combined multi-wave solutions for (2+1)-dimensional variable coefficients KdV equation. Nonlinear Dyn. 2018;93(2):733–40. 10.1007/s11071-018-4222-1Search in Google Scholar

[11] Park C, Nuruddeen RI, Ali KK, Muhammad L, Osman MS, Baleanu D. Novel hyperbolic and exponential ansatz methods to the fractional fifth-order Korteweg-de Vries equations. Adv Differ Equ. 2020;2020(1):1–12. 10.1186/s13662-020-03087-wSearch in Google Scholar

[12] Nisar KS, Ilhan OA, Abdulazeez ST, Manafian J, Mohammed SA, Osman MS. Novel multiple soliton solutions for some nonlinear PDEs via multiple Exp-function method. Results Phys. 2021;21:103769. 10.1016/j.rinp.2020.103769Search in Google Scholar

[13] Djennadi S, Shawagfeh N, Osman MS, Gómez-Aguilar JF, Arqub OA. The Tikhonov regularization method for the inverse source problem of time fractional heat equation in the view of ABC-fractional technique. Phys Scr. 2021;96(9):094006. 10.1088/1402-4896/ac0867Search in Google Scholar

[14] Ismael HF, Bulut H, Par C, Osman MS. M-lump, N-soliton solutions, and the collision phenomena for the (2+1)-dimensional Date-Jimbo-Kashiwara-Miwa equation. Results Phys. 2020;19:103329. 10.1016/j.rinp.2020.103329Search in Google Scholar

[15] Shahoot AM, Alurrfi KAE, Hassan IM, Almsri AM. Solitons and other exact solutions for two nonlinear PDEs in mathematical physics using the generalized projective Riccati equations method. Adv Math Phys. 2018;2018:6870310. 10.1155/2018/6870310Search in Google Scholar

[16] Dai CQ, Zhang JF. New types of interactions based on variable separation solutions via the general projective Riccati equation method. Rev Math Phys. 2007;19(2):195–226. 10.1142/S0129055X07002948Search in Google Scholar

[17] Akram G, Sadaf M, Khan MAU. Abundant optical solitons for Lakshmanan-Porsezian-Daniel model by the modified auxiliary equation method. Optik. 2022;251:168163. 10.1016/j.ijleo.2021.168163Search in Google Scholar

[18] Akram G, Sadaf M, Khan MAU. Soliton solutions of Lakshmanan-Porsezian-Daniel model using modified auxiliary equation method with parabolic and anti-cubic law of nonlinearities. Optik. 2022;252:168372. 10.1016/j.ijleo.2021.168372Search in Google Scholar

[19] Akram G, Sadaf M, Zainab I. New graphical observations for Kdv equation and KdV-Burgers equation using modified auxiliary equation method. Modern Phys Lett B. 2022;36(01):2150520. 10.1142/S0217984921505205Search in Google Scholar

[20] Osman MS. Multi-soliton rational solutions for some nonlinear evolution equations. Open Phys. 2016;14(1):26–36. 10.1515/phys-2015-0056Search in Google Scholar

[21] Osman MS, Machado JAT. The dynamical behavior of mixed-type soliton solutions described by (2+1)-dimensional Bogoyavlensky-Konopelchenko equation with variable coefficients. J Electromagnetic Waves Appl. 2018;32(11):1457–64. 10.1080/09205071.2018.1445039Search in Google Scholar

[22] Osman MS. Analytical study of rational and double-soliton rational solutions governed by the KdV-Sawada-Kotera-Ramani equation with variable coefficients. Nonlinear Dyn. 2017;89(3):2283–9. 10.1007/s11071-017-3586-ySearch in Google Scholar

[23] Li X, Wang M. A sub-ODE method for finding exact solutions of a generalized KdV-mKdV equation with high-order nonlinear terms. Phys Lett A. 2007;361(1–2):115–8. 10.1016/j.physleta.2006.09.022Search in Google Scholar

[24] Mohamad MNB. Exact solutions to the combined KdV and mKdV equation. Math Methods Appl Sci. 1992;15(2):73–8. 10.1002/mma.1670150202Search in Google Scholar

[25] Wadati M. Wave propagation in nonlinear lattice. I. J Phys Soc Japan. 1975;38(3):673–80. 10.1143/JPSJ.38.673Search in Google Scholar

Received: 2023-02-25

Revised: 2023-07-27

Accepted: 2023-08-15

Published Online: 2024-04-16

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

Articles in the same Issue

https://doi.org/10.1515/nleng-2022-0318

Keywords for this article

generalized Riccati equation method; modified auxiliary equation method; generalized unified method; exact solutions

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