# New Optical Solitons in Bragg Grating Fibers for the Nonlinear Coupled (2+1)-Dimensional Kundu-Mukherjee-Naskar System via Complete Discrimination System Method

Authors: Chen Peng; Furong Zhang; Hongwei Zhao; Zhao Li
Journal: Advances in Mathematical Physics
 (2022)

---

## Abstract

In this paper, the coupled Kundu-Mukherjee-Naskar (KMN) model in Bragg grating fibers is considered to retrieve some new optical soliton solutions in (2+1) dimensions. Plenty of new exact solutions, including rational function solutions and triangle function solutions, in addition to the Jacobian elliptic function solutions, are obtained by using a complete discrimination system method. The 3D-surface plots, 2D-shape plots, and corresponding 2D contour plots of some obtained solutions are drawn, which provides a visualized structures and propagation of solitons. The novel results are new and show the effectiveness of the proposed method.


---

## Body

## 1. Introduction

The Kundu-Mukherjee-Naskar (KMN) equation was first proposed in 2014 to address rogue waves (RWs) in the ocean [1], which is usually denotes as
(1)iqt+aqxy+ibqqx∗−q∗qx=0.In recent years, many mathematicians and physicists have devoted themselves to the research and discussion of this equation. In [2], Qiu et al. considered this model to obtain first-order oceanic rogue wave solution via the Darboux transformation. In [3], Singh et al. obtained higher-dimensional nonlinear wave solutions and study the dynamics of deep water oceanic RWs. In [4], Ekici et al. indicated that the KMN equation is applicable to study the dynamics of soliton propagation through optical fibers in (2+1) dimensions on the bases of the fact that RWs are observed in a crystal fiber [5]. Note that the use of special methods constructing exact solutions of nonlinear differential models (see [6–13]) is a main research area of nonlinear optical science, the optical solitons in KMN equation have been addressed by broad researchers to recover the exact solutions by applying many effective methods including Kudryashov’s approach method [14, 15], the tanh function method [16], the new auxiliary equation method [17], the new extended direct algebraic method [18, 19], the method of undetermined coefficients and Lie symmetry [20, 21], the trial equation technique [22], the functional variable method [23], and the modified simple equation approach technique [24]. It is worth mentioning that Yldrm [25] also used the modified simple equation approach technique to discuss a new model of coupled KMN equations in birefringent fibers. Then, Zayed et al. [26] first proposed the coupled KMN model in the Bragg gratings fibers.In this study, we consider the nonlinear coupled (2+1)-dimensional KMN model in the Bragg grating fibers [26]:
(2)iψt+a1ϕxy+ib1ψ2+c1ϕ2ψx∗−d1ψ2+e1ϕ2ψx+iα1ψx+β1ϕ+σ1ψ∗ϕ2=0,iϕt+a2ψxy+ib2ϕ2+c2ψ2ϕx∗−d2ϕ2+e2ψ2ϕx+iα2ϕx+β2ψ+σ2ϕ∗ψ2=0,where ψx,y,t and ϕx,y,t are complex-valued functions that represent the wave profiles, while aj, bj, cj, dj, ej, αj, βj, and σjj=1;2 are real-valued constants. The parameters ajj=1;2 are the coefficients of dispersion terms. The parameters bj, cj, dj, and ej are the coefficients of nonlinearity. The parameters αj, βj, and σj give the intermodal dispersions, the detuning parameters, and the four wave mixing parameters, respectively. In the same work [26], the modified and addendum Kudryashov’s method are used to obtain optical solitons for model (2).Recently, the complete discrimination system for the polynomial method was first proposed by Liu [27] as it is modified form of the discrimination system for high degree polynomial and provides a bit wider range of exact solutions than the famous methods mentioned above. Many authors have solved a lot of models in mathematics, physics, engineering, fluid mechanics, plasma, optical fibers, and other areas of science (see [28–33]). The aim of this paper is to extract the new exact optical solutions of the model (2) after utilizing Liu’s method.The rest of the paper is organized as follows: In Section2, a glance of discrimination system for polynomial method is given. In Section 3, after taking travelling wave transformation upon (2), the model is reduced to ordinary differential equations (ODEs). Implementing the complete discrimination system method to the ODEs, the exact solutions are obtained. In Section 4, the focus is on the graphical representation for the obtained solutions. We fix parameters to draw the 3D-surface plots, 2D-shape plots, and the corresponding 2D contour plots of some obtained solutions. Finally, the comparison of the obtained results is discussed and conclusions are illustrated in Section 5.


## 2. Outline of the Complete Discrimination System Method

We consider a nonlinear differential equation takes the form as follows:(3)Gp,pt,px,pxt,pxx,pxxx,⋯=0,where p=pt,x is an unknown function and G is a polynomial of all the derivatives of p=pt,x.By applying the classical complex traveling wave transformationpx,t=uξeiϕ, we can convert Equation (3) into an ordinary differential equation (ODE), which can be written as
(4)Fu,u′,u′′,⋯=0,where F is a polynomial of u and its derivatives, notation “′” denotes the derivative with respect to ξ.Next, Equation (4) can be reduced to
(5)±ξ−ξ0=∫duFu,where ξ0 is an integral constant and Fu denotes a n degree polynomial. According to the complete discriminant system method, we can retrieve all types of single wave solutions of Equation (2) from Equation (5) by classifying the roots for polynomial Fu.


## 3. Applications to the Model

Firstly, we assume Equation (2) have solutions in the form:
(6)ψx,y,t=uξeiζx,y,t,ϕx,y,t=vξeiζx,y,t,ξ=K1x+K2y−νt,ζx,y,t=−k1x−k2y+ωt+θ,where uξ and vξ are the functions stand for the profile for optical pulse, and ζx,y,t describes the phase portion. Here, the parameters K1 and K2 in the amplitude component stand for the direct cosines of the solitons along the x- and y-directions, respectively, and ν denotes the soltion velocity. The parameters k1 and k2 in the amplitude component signify the frequencies of the solitons along the x- and y-directions, respectively, while ω is the wave number and θ is the phase constant.Inserting (6) into (2) and decomposing into the real and imaginary parts, we get the real parts for the uξ and vξ as
(7)a1K1K2v′′ξ+β1−a1k1k2vξ+α1k1−ωuξ−b1+d1k1u3ξ+σ1−c1+e1k1uξv2ξ=0,(8)a2K1K2u′′ξ+β2−a2k1k2uξ+α2k1−ωvξ−b2+d2k1v3ξ+σ2−c2−e2k1u2ξvξ=0,while the imaginary parts are
(9)α1K1−νu′ξ−a1k2K1+K2k1v′ξ+K1b1−d1u2ξu′ξ+K1c1−e1v2ξu′ξ=0,(10)α2K1−νv′ξ−a2k1K2+K1k2u′ξ+K1b2−d2v2ξv′ξ+K1c2−e2u2ξv′ξ=0.In the view of physical reality, we suppose that(11)vξ=Auξ,where A≠0 is a constant.Then, implementing the condition (11) in (7)–(8), the real parts are rewritten as
(12)a1K1K2Au′′ξ+Aβ1−a1k1k2+α1k1−ωuξ+A2σ1−c1+e1k1−b1+d1k1u3ξ=0,(13)a2K1K2u′′ξ+β2−a2k1k2+Aα2k1−ωuξ+Aσ2−c2+e2k1−A3b2+d2k1u3ξ=0.Making Equations (12) and (13) equal for convenient, the coefficients satisfy the following results:
(14)a1A=a2,(15)Aβ1−α2k1+ω=β2−α1k1+ω,(16)A2σ1−c1+e1k1−b1+d1k1=Aσ2−c2+e2k1−A3b2+d2k1.Putting (11) into (9) and (10), the imaginary parts become
(17)α1K1−ν−a1Ak2K1+K2k1u′ξ+K1b1−d1+A2c1−e1u2ξu′ξ=0,(18)Aα2K2+v−a2k1K2+K1k2u′ξ+AK2A2b2−d2+c2−e2u2ξu′ξ=0.Taking into account the linear independence of (17) and (18) and making the coefficients to zero, we obtain the constrain conditions:
(19)α1K1−ν−Aa1k2K1+K2k1=0,Aα2K1−ν−a2k1K2+K1k2=0,(20)b1−d1+A2c1−e1=0,A2b2−d2+c2−e2=0.Implementing (14) in (19), the velocity of the soliton is obtained as
(21)ν=α1K1−a2k1K2+K1k2=α2K1−a1k1K2+K1k2.Multiplyingu′ on both sides of (12) and integrating it on ξ, we get
(22)u′ξ2=b1+d1k1−A2σ1−c1+e1k12a1K1K2Au4ξ+ω−α1k1−Aβ1−a1k1k2a1K1K2Au2ξ+C0,where C0 is an arbitrary constant.For obtaining exact solutions, the following transformations are selected:(23)u=±2b1+d1k1−2A2σ1−c1+e1k1a1K1K2A−1/3w,ξ1=2b1+d1k1−2A2σ1−c1+e1k1a1K1K2A1/3ξ.Inserting (23) into (22), we yield
(24)wξ1′2=w3ξ1+pw2ξ1+qwξ1,where q is the an arbitrary constant and coefficient p has the forms:
(25)p=4ω−α1k1−Aβ1−a1k1k2a1K1K2A2b1+d1k1−2A2σ1−c1+e1k1a1K1K2A−2/3.It is easy to reformulate Equation (24) with an integral representation as
(26)±ξ1−ξ0=∫dwww2+pw+q,where ξ0 is the integration constant.Note that the signΔ=p2−4q and p comprise the complete discriminant system for the polynomial Fw=w3+pw2+qw, and we retrieve the exact solutions of (24) by applying the complete discrimination system method in the paragraph below.Case 1.
Δ=0. Then, we have Fw=ww+p/22. When w>0, the exact traveling wave solutions for Equation (2) are recovered as below:
If−p/2>0, the exact solutions for Equation (2) are
(27)ψ1x,y,t=±−p22b1+d1k1−2A2σ1−c1+e1k1a1K1K2A−1/6×ei−k1x−k2y+ωt+θ×tanh−p/222b1+d1k1−2A2σ1−c1+e1k1a1K1K2A1/3K1x+K2y−νt−ξ0,(28)ϕ1x,y,t=±A−p22b1+d1k1−2A2σ1−c1+e1k1a1K1K2A−1/6×ei−k1x−k2y+ωt+θ×tanh−p/222b1+d1k1−2A2σ1−c1+e1k1a1K1K2A1/3K1x+K2y−νt−ξ0,(29)ψ2x,y,t=±−p22b1+d1k1−2A2σ1−c1+e1k1a1K1K2A−1/6×ei−k1x−k2y+ωt+θ×coth−p/222b1+d1k1−2A2σ1−c1+e1k1a1K1K2A1/3K1x+K2y−νt−ξ0,(30)ϕ2x,y,t=±A−p22b1+d1k1−2A2σ1−c1+e1k1a1K1K2A−1/6×ei−k1x−k2y+ωt+θ×coth−p/222b1+d1k1−2A2σ1−c1+e1k1a1K1K2A1/3K1x+K2y−νt−ξ0.If−p/2<0, the solution is
(31)ψ3x,y,t=±p22b1+d1k1−2A2σ1−c1+e1k1a1K1K2A−1/6×ei−k1x−k2y+ωt+θ×tanp/222b1+d1k1−2A2σ1−c1+e1k1a1K1K2A1/3K1x+K2y−νt−ξ0,(32)ϕ3x,y,t=±Ap22b1+d1k1−2A2σ1−c1+e1k1a1K1K2A−1/6×ei−k1x−k2y+ωt+θ×tanp/222b1+d1k1−2A2σ1−c1+e1k1a1K1K2A1/3K1x+K2y−νt−ξ0.If−p/2=0, we get
(33)ψ4x,y,t=±22b1+d1k1−2A2σ1−c1+e1k1a1K1K2A−1/6×ei−k1x−k2y+ωt+θ×2b1+d1k1−2A2σ1−c1+e1k1a1K1K2A1/3K1x+K2y−νt−ξ0−1,ϕ4x,y,t=±2A2b1+d1k1−2A2σ1−c1+e1k1a1K1K2A−1/6×ei−k1x−k2y+ωt+θ×2b1+d1k1−2A2σ1−c1+e1k1a1K1K2A1/3K1x+K2y−νt−ξ0−1.Case 2.
Δ>0, q=0. Then, we have Fw=w2w+p. When w>−p, the exact traveling wave solutions for Equation (2) are obtained as follows.
If0>−p, we have
(34)ψ5x,y,t=±−2b1+d1k1−2A2σ1−c1+e1k1a1K1K2A−1/6×ei−k1x−k2y+ωt+θ×p−ptanh2p22b1+d1k1−2A2σ1−c1+e1k1a1K1K2A1/3K1x+K2y−νt−ξ01/2,(35)ϕ5x,y,t=±A−2b1+d1k1−2A2σ1−c1+e1k1a1K1K2A−1/6×ei−k1x−k2y+ωt+θ×p−ptanh2p22b1+d1k1−2A2σ1−c1+e1k1a1K1K2A1/3K1x+K2y−νt−ξ01/2,(36)ψ6x,y,t=±2b1+d1k1−2A2σ1−c1+e1k1a1K1K2A−1/6×ei−k1x−k2y+ωt+θ×pcoth2p22b1+d1k1−2A2σ1−c1+e1k1a1K1K2A1/3K1x+K2y−νt−ξ0−p1/2,(37)ϕ6x,y,t=±A2b1+d1k1−2A2σ1−c1+e1k1a1K1K2A−1/6×ei−k1x−k2y+ωt+θ×pcoth2p22b1+d1k1−2A2σ1−c1+e1k1a1K1K2A1/3K1x+K2y−νt−ξ0−p1/2.If0<−p, we have
(38)ψ7x,y,t=±2b1+d1k1−2A2σ1−c1+e1k1a1K1K2A−1/6−p×ei−k1x−k2y+ωt+θ×sec−p22b1+d1k1−2A2σ1−c1+e1k1a1K1K2A1/3K1x+K2y−νt−ξ0,(39)ϕ7x,y,t=±A2b1+d1k1−2A2σ1−c1+e1k1a1K1K2A−1/6−p×ei−k1x−k2y+ωt+θ×sec−p22b1+d1k1−2A2σ1−c1+e1k1a1K1K2A1/3K1x+K2y−νt−ξ0.Case 3.
Δ>0, q≠0, that is, Fw=ww−λ1w−λ2. We note that, in this case, λ1≠λ2≠0.
If0<λ1<λ2, when 0<w<λ1, the exact solution for Equation (2) takes the form
(40)ψ8x,y,t=±λ12b1+d1k1−2A2σ1−c1+e1k1a1K1K2A−1/6×ei−k1x−k2y+ωt+θ×snλ222b1+d1k1−2A2σ1−c1+e1k1a1K1K2A1/3K1x+K2y−νt−ξ0,m,ϕ8x,y,t=±Aλ12b1+d1k1−2A2σ1−c1+e1k1a1K1K2A−1/6×ei−k1x−k2y+ωt+θ×snλ222b1+d1k1−2A2σ1−c1+e1k1a1K1K2A1/3K1x+K2y−νt−ξ0,m,when λ2<w, the exact solution for Equation (2) takes the form
(41)ψ9x,y,t=±2b1+d1k1−2A2σ1−c1+e1k1a1K1K2A−1/6×ei−k1x−k2y+ωt+θ×λ2−λ1sn2λ2/22b1+d1k1−2A2σ1−c1+e1k1/a1K1K2A1/3K1x+K2y−νt−ξ0,mcn2λ2/22b1+d1k1−2A2σ1−c1+e1k1/a1K1K2A1/3K1x+K2y−νt−ξ0,m1/2,ϕ9x,y,t=±A2b1+d1k1−2A2σ1−c1+e1k1a1K1K2A−1/6×ei−k1x−k2y+ωt+θ×λ2−λ1sn2λ2/22b1+d1k1−2A2σ1−c1+e1k1/a1K1K2A1/3K1x+K2y−νt−ξ0,mcn2λ2/22b1+d1k1−2A2σ1−c1+e1k1/a1K1K2A1/3K1x+K2y−νt−ξ0,m1/2,where m2=λ1/λ2.Ifλ1<0<λ2, when λ1<w<0, the exact solution for Equation (2) takes the form
(42)ψ10=±−λ1−2b1+d1k1−2A2σ1−c1+e1k1a1K1K2A−1/6×ei−k1x−k2y+ωt+θ×−1+sn2λ2−λ122b1+d1k1−2A2σ1−c1+e1k1a1K1K2A1/3K1x+K2y−νt−ξ0,m1/2,ϕ10=±A−λ1−2b1+d1k1−2A2σ1−c1+e1k1a1K1K2A−1/6×ei−k1x−k2y+ωt+θ×−1+sn2λ2−λ122b1+d1k1−2A2σ1−c1+e1k1a1K1K2A1/3K1x+K2y−νt−ξ0,m1/2,when λ2<w, the exact solution for Equation (2) takes the form
(43)ψ11=±λ22b1+d1k1−2A2σ1−c1+e1k1a1K1K2A−1/6×ei−k1x−k2y+ωt+θ×cnλ2−λ122b1+d1k1−2A2σ1−c1+e1k1a1K1K2A1/3K1x+K2y−νt−ξ0,m−1,ϕ11=±A−λ22b1+d1k1−2A2σ1−c1+e1k1a1K1K2A−1/6×ei−k1x−k2y+ωt+θ×cnλ2−λ122b1+d1k1−2A2σ1−c1+e1k1a1K1K2A1/3K1x+K2y−νt−ξ0,m−1,where m2=−λ1/λ2−λ1.Ifλ1<λ2<0, when λ1<w<λ2, the exact solution for Equation (2) takes the form
(44)ψ12=±2b1+d1k1−2A2σ1−c1+e1k1−a1K1K2A−1/6×ei−k1x−k2y+ωt+θ×λ1−λ2−λ1sn2−λ122b1+d1k1−2A2σ1−c1+e1k1a1K1K2A1/3K1x+K2y−νt−ξ0,m1/2,ϕ12=±A2b1+d1k1−2A2σ1−c1+e1k1−a1K1K2A−1/6×ei−k1x−k2y+ωt+θ×λ1−λ2−λ1sn2−λ122b1+d1k1−2A2σ1−c1+e1k1a1K1K2A1/3K1x+K2y−νt−ξ0,m1/2,when 0<w, the exact solution for Equation (2) takes the form
(45)ψ13x,y,t=±−λ22b1+d1k1−2A2σ1−c1+e1k1a1K1K2A−1/6×ei−k1x−k2y+ωt+θ×sn−λ1/22b1+d1k1−2A2σ1−c1+e1k1/a1K1K2A1/3K1x+K2y−νt−ξ0,mcn−λ1/22b1+d1k1−2A2σ1−c1+e1k1/a1K1K2A1/3K1x+K2y−νt−ξ0,m,ϕ13x,y,t=±A−λ22b1+d1k1−2A2σ1−c1+e1k1a1K1K2A−1/6×ei−k1x−k2y+ωt+θ×sn−λ1/22b1+d1k1−2A2σ1−c1+e1k1/a1K1K2A1/3K1x+K2y−νt−ξ0,mcn−λ1/22b1+d1k1−2A2σ1−c1+e1k1/a1K1K2A1/3K1x+K2y−νt−ξ0,m,where m2=λ2−λ1/−λ1.Case 4.
Δ<0. When w>0, we obtain the exact traveling wave solutions for Equation (2) as
(46)ψ14x,y,tψ14x,y,t=±q4×2b1+d1k1−2A2σ1−c1+e1k1a1K1K2A−1/6×ei−k1x−k2y+ωt+θ×1−cnq42b1+d1k1−2A2σ1−c1+e1k1/a1K1K2A1/3K1x+K2y−νt−ξ0,m1+cnq42b1+d1k1−2A2σ1−c1+e1k1/a1K1K2A1/3K1x+K2y−νt−ξ0,m1/2,ϕ14x,y,t=±Aq4×2b1+d1k1−2A2σ1−c1+e1k1a1K1K2A−1/6×ei−k1x−k2y+ωt+θ×1−cnq42b1+d1k1−2A2σ1−c1+e1k1/a1K1K2A1/3K1x+K2y−νt−ξ0,m1+cnq42b1+d1k1−2A2σ1−c1+e1k1/a1K1K2A1/3K1x+K2y−νt−ξ0,m1/2,where m2=2q−p/4q.


## 4. Graphical Representation

In this section, we graphically illustrate the amplitude profile of some retrieved solutions of four types by using Maple software. Graphs 1-4 show the 2D, 3D, and corresponding contour plots for the solutionsψ1, ψ3, ψ5, and ψ7, respectively, by taking suitable choice of free parameters satisfying all the constrict conditions (14)–(16) and (19)–(20). Figure 1 shows the 2D and 3D views and contour plot of kink soliton solution of (27) by taking suitable parameters. Figures 2 and 3 elaborate both cases of singular periodic solutions of (31) and (38) in 2D and 3D shapes and contour plot. The 2D and 3D views and contour plot of the bright soliton solution of (34) is depicted in Figure 4. Figures 1–4 express an insight view of the structures for the solutions of (2) which give us a better understanding of the propagation and mechanism for the origin system.Figure 1
The graph shows solutionψ1 by assuming a1=0.1, a2=0.2, b1=0.24, b2=0.03, c1=0.03, c2=0.06, d1=0.16, d2=0.02, e1=0.01, e2=0.02, α1=2.08, α2=2.04, β1=0.002, β2=0.014, σ1=0.0256, σ2=0.0512, y=1, θ=0, and ξ0=0. Letting K1=0.2, K2=0.3, k1=0.2, and k2=0.1, then ν=0.4, ω=0.41, p=−2, and q=1.
(a)
3D surface plot of (27)(b)
2D shape of (27) at t=5 along the x-axis(c)
2D contour plot ofu1Figure 2
The graph shows solutionψ3 by assuming a1=0.1, a2=0.2, b1=0.24, b2=0.03, c1=0.03, c2=0.06, d1=0.16, d2=0.02, e1=0.01, e2=0.02, α1=2.08, α2=2.04, β1=0.002, β2=−0.768, σ1=0.0256, σ2=0.0512, y=1, θ=0, and ξ0=0. Letting K1=0.2, K2=0.3, k1=0.2, and k2=0.1, then ν=0.4, ω=−0.372, p=4, and q=1.
(a)
3D surface plot of (31)(b)
2D shape of (31) at t=1 along x-axis(c)
2D contour plot ofu3Figure 3
The graph shows solutionψ7 by assuming a1=0.1, a2=0.2, b1=0.24, b2=0.03, c1=0.03, c2=0.06, d1=0.16, d2=0.02, e1=0.01, e2=0.02, α1=−1.92, α2=−1.96, β1=0.002, β2=0.006, σ1=0.0256, σ2=0.0512, y=3, θ=0, and ξ0=0. Letting K1=0.2, K2=0.3, k1=0.2, and k2=0.1, then ν=−0.4, ω=−0.39, p=−2, and q=0.
(a)
3D surface plot of (38)(b)
2D shape of (38) at t=1 along x-axis(c)
2D contour plot ofu7Figure 4
The graph shows solutionψ5 by assuming a1=0.1, a2=0.2, b1=0.24, b2=0.03, c1=0.03, c2=0.06, d1=0.16, d2=0.02, e1=0.01, e2=0.02, α1=−1.92, α2=−1.96, β1=0.002, β2=0.032, σ1=0.0295, σ2=0.059, y=1, θ=0, and ξ0=0. Letting K1=0.2, K2=0.3, k1=0.2, and k2=0.1, then ν=−0.4, ω=−0.372, p=4, and q=0.
(a)
3D surface plot of (34)(b)
2D shape of (34) at t=5 along x-axis(c)
2D contour plot ofu5


## 5. Conclusion

The governing model in Bragg grating fibers for the coupled Kundu-Mukherjee-Naskar model was examined on purpose of obtaining new optical soliton solutions. The complete discrimination system for the polynomial method was adopted to uncover the rational function solutions, triangle function solutions, and the Jacobian elliptic function solutions of this model. All of the derived solutions are guaranteed by putting them back into the original equation. Many other structures can be obtained by choosing free parameters. Comparing the results with the works done in [26], some of our obtained solutions are in good agreement with the published results which is presented in Table 1. The solutions (31) and (32) and (38) and (39) were not obtained by Zayed et al. [26]. The listed solutions in Case 3 and 4 are reported for the first time and can hardly be obtained by other methods. As a rather newly proposed system, there is a lot of scope to extend the horizon in this context. For this reason, additional techniques such as the Lie group, the new extended direct algebraic method, and the new auxiliary equation method will be considered on this model. These details shall be revealed, respectively, with time.Table 1
Comparison between Zayed et al. ([26]) solutions and our solutions.
Zayed et al. (2022) solutionsOur solutions(i) IfL1=1/2, L2=−1, ε=1 then solutions(i) Ifq=1, p=−2, then we obtain(35) reduced touξ=−1/2tanh1/2ξ−ξ0u1ξ=±12tanh12ξ−ξ0(ii) IfL1=1/2, L2=−1, ε=−1 then solutions(ii) Ifq=1, p=−2, then we obtain(37) Reduced touξ=−1/2coth1/2ξ−ξ0u2ξ=±12coth12ξ−ξ0



---

*Source: 8184270-2022-06-11.xml*