Research article

Analysis of optical solitons solutions of two nonlinear models using analytical technique

  • Received: 11 May 2021 Accepted: 23 August 2021 Published: 15 September 2021
  • MSC : 35Q51, 35Q53

  • Looking for the exact solutions in the form of optical solitons of nonlinear partial differential equations has become very famous to analyze the core structures of physical phenomena. In this paper, we have constructed some various type of optical solitons solutions for the Kaup-Newell equation (KNE) and Biswas-Arshad equation (BAE) via the generalized Kudryashov method (GKM). The conquered solutions help to understand the dynamic behavior of different physical phenomena. These solutions are specific, novel, correct and may be beneficial for edifying precise nonlinear physical phenomena in nonlinear dynamical schemes. Graphical recreations for some of the acquired solutions are offered.

    Citation: Naeem Ullah, Muhammad Imran Asjad, Azhar Iqbal, Hamood Ur Rehman, Ahmad Hassan, Tuan Nguyen Gia. Analysis of optical solitons solutions of two nonlinear models using analytical technique[J]. AIMS Mathematics, 2021, 6(12): 13258-13271. doi: 10.3934/math.2021767

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  • Looking for the exact solutions in the form of optical solitons of nonlinear partial differential equations has become very famous to analyze the core structures of physical phenomena. In this paper, we have constructed some various type of optical solitons solutions for the Kaup-Newell equation (KNE) and Biswas-Arshad equation (BAE) via the generalized Kudryashov method (GKM). The conquered solutions help to understand the dynamic behavior of different physical phenomena. These solutions are specific, novel, correct and may be beneficial for edifying precise nonlinear physical phenomena in nonlinear dynamical schemes. Graphical recreations for some of the acquired solutions are offered.



    Recently, nonlinear evolution equations (NLEEs) has been developed as specific modules of the class of partial differential equations (PDEs). It is distinguished that investigating exact solutions for NLEEs, via many dissimilar methods shows an active part in mathematical physics and has become exciting and rich zones of research analysis for physicist and mathematicians. Lots of significant dynamic processes and phenomena in biology, chemistry, mechanics and physics can be expressed by nonlinear partial differential equations (NLPDEs). In NLEEs, nonlinear wave phenomena of diffusion, dispersion, reaction, convection and dissipation are very important. It is necessary to define exact traveling solutions for these nonlinear equations to analyze various properties of these equations. Nowadays, NLEEs has become a significant area of research. Mostly, the existence of soliton solutions for NLEEs is of much important because of their widely applications in various areas of mathematical biology, chaos, neural physics, optical fibers and solid state physics etc.

    Solitons are considered in the fields as optical communication, plasma, medical imaging, super continuum generation and nonlinear optics etc. They can transmit without changing their amplitude, velocity and wave form for a long distance. Optical soliton forms the excellent transporter minutes in the telecommunication engineering. Nowadays, some methods have been established for discovering exact traveling and solitary wave solutions of NLEEs. Various influential methods for instance, auxiliary equation method [1,2], homogeneous balance method [3,4], the Exp-function method [5,6], the tanh-function method [7], Darboux transformation method [8,9], the modified extended tanh-function method [10], the first integral method [11,12], Jacobi elliptic function method [13,14], the modified simple equation method [15,16,17], the exp(-F(x))-expansion method [18], the (G/G)-expansion method [19,20,21,22,23,24,25,26], the variational iteration method [27] the homotopy perturbation method [28,29,30,31,32], the F-expansion method [33,34,35] and many more [36,37,38,39,40,41]. Many models are existing to report this dynamic in the structure of optical fibers. The Schrodinger equation, the important model in submicroscopic phenomena and developed a fundamental importance to quantum mechanics. Such model denotes to the form of wave functions that manage the motion of small particles and classifies how these waves are transformed by external impacts. It has been measured and considered in several designs.

    In optical fibers, most of these models are frequently stated in the time domain, and when fields at dissimilar frequencies spread through the fiber the common practice is also to transcribe a distance equation for each field component. The nonlinear transformation of dielectric of the fiber termed as the Kerr effect is applied to neutralize the dispersion effect, in this state, the optical pulse might lean to form a steady nonlinear pulse known as an optical soliton. The bit rate of transmission is restricted by the dispersion of the fiber material. Soliton transmission is an area of huge interest since of the wide applications in ultrafast signal routing systems, transcontinental and short-light-pulse telecommunication [42,43,44,45]. In this work, we used the generalized Kudryashov method to construct the exact traveling wave solutions for the Kaup-Newell equation (KNE) and Biswas-Arshad equation (BAE). The KNE is a significant model with many applications in optical fibers. The dynamics of solitons in optical tools is observed as an important arena of research in nonlinear optics that has added much attention in the past few decades [46,47,48,49]. The transmission of waves in optical tools with Kerr dispersion rests important in construction to the so-called time evolution equations [50,51,52]. The three models that invent from the Nonlinear Schrodinger equation (NLSE) which are termed as the Derivative Nonlinear Schrodinger Equations (DNLSE) are classified into three classes: I, II, and III. Connected in this study is the DNLSE-I which is in its place known as the KNE. This class will be the focal point of the current study and a lot of inquiries have correctly been approved in the literature. Recently, Biswas and Arshad [53] constructed a model from the NLSE known as Biswas-Arshed equation (BAE). The BAE is one of the important models in the telecommunications industry. The extreme remarkable story of this model is that the self-phase modulation is ignored and likewise GVD is negligibly slight. The plus point of this method is that it offers further novel exact solutions in optical solitons form.

    The draft of this paper is organized like this. Section 2, contained the description of the generalized Kudryashov method. In section 3, application of GKM for KNE is presented. Section 4 presents the application of GKM for BAE. Section 5 contains results and discussion. Conclusion of the paper is discussed in section 6.

    The steps of GKM [54] are as follows

    Let NLEE in the form

    W(q,qx,qt,qxt,qtt,qxx,)=0, (2.1)

    where q=q(x,t) is a function.

    Step 1. Applying the following wave transformation

    q(x,t)=g(η),η=(xvt), (2.2)

    into (2.1), so (2.1) converts to nonlinear ODE in the form

    H(q,q,q,q,)=0, (2.3)

    here, q is a function of η and q=dq/dη and v is the wave-speed.

    Step 2. Let (2.3) has the solution in form

    g(η)=Ni=0aiSi(η)Mj=0bjSj(η), (2.4)

    where ai, bj are constants and (i0N), (j0M) such that aN0, bM0.

    S(η)=11+Aeη, (2.5)

    is the solution in the form

    dS(η)dη=S2(η)S(η), (2.6)

    where A is constant.

    Step 3. Using balancing rule in (2.3) to obtain the values of N and M.

    Step 4. Utilizing (2.4) and (2.6) into (2.3), we get an expression in Si, where (i=0,1,2,3,4,). Then collecting all the coefficients of Si with same power(i) and equating to zero, we get a system of alebraic equations in all constant terms. This system of algebraic equations can be solved by Maple to unknown parameters.

    The governing equation [55] is given as:

    iqt+aqxx+ib(|q|2q)x=0. (3.1)

    Here, q(x,t) is a complex valued function, indicates the wave pfofile and rests on variables, space x and time t. It includes the non-Kerr dispersion, evolution and and GVD terms. Also, a is the coefficient of GVD and b is the coefficient of self-steepening term.

    Suppose (3.1) has the following solution

    q(x,t)=g(η)eiϕ(x,t), (3.2)

    where

    η=αxct,ϕ(x,t)=κx+ωt. (3.3)

    Here g(η), κ, c and ω are the amplitude, frequency, speed and wave number of the pulse, respectively. Putting (3.2) into (3.1), and splitting into imaginary and real parts.

    The imaginary part has the form

    cg2aκαg+3bαg2g=0. (3.4)

    We can get easily the value of c as under

    c=2ακa,

    and the constraint condition as under

    3αbg2=0.

    The real part has the form

    aα2g+κbg3(aκ2+ω)g=0. (3.5)

    Now, balancing the g and non-linear term g3 in (3.5), we get N=M+1. So for M=1, we get N=2.

    The solution of (3.5) by generalized Kudryashov method as given in (2.4), reduces to the form

    g(η)=a0+a1S(η)+a2S2(η)b0+b1S(η), (3.6)

    a0, a1, a2, b0 and b1 are constants. Subtitling the (3.6) into (3.5) and also applying (2.6), we get an expression in S(η). Collecting the coefficients of same power of Si and equating to zero, the system of equations is obtained, as follows.

    {bκa30+(aκ2ω)a0b20=0,3bκa20a1+(aκ2ω)a1b20+aα2a1b20+2(aκ2ω)a0b0b1aα2a0b0b1=0,3bκa0a21+3bκa20a23aα2a1b20+(aκ2ω)a2b20+4aα2a2b20+3aα2a0b0b1+2(aκ2ω)a1b0b1aα2a1b0b1+(aκ2ω)a0b21+aα2a0b21=0,bκa21+6bκa0a1a2+2aα2a1b2010aα2a2b202aα2a1b0b1+aα2a1b0b1+2(aκ2ω)a2b0b1+3aα2a2b0b1aα2a0b21+(aκ2ω)a1b21=0,3bκa21a2+3bκa0a22+6aα2a2b209aα2a2b0b1+(aκ2ω)a2b21+aα2a2b21=0,3bκa1a22+6aα2a2b0b13aα2a2b21=0,bκa32+2aα2a2b21=0. (3.7)

    By solving the above system, we get various types of solutions. These solutions are deliberated below.

    Case 1.

    a0=0,a1=a1,a2=a1,b0=12b1,b1=b1,
    a=a,κ=2aα2b21a21b,ω=aα2(4a2α2b41a41b2)a41b2.

    Case 1 corresponds the following solution for Kaup-Newell equation

    q(x,t)=(a1(1+Aeαxct)a1(1+Aeαxct)212b1+b1(1+Aeαxct))×eι(2aα2b21a21bxaα2(4a2α2b41a41b2)a41b2t). (3.8)

    Case 2.

    a0=a0,a1=(b1+2b0)a0b0,a2=0,b0=b0,b1=b1.
    a=a,κ=aα2b202ba20,ω=aα2(2b2a40+b40a2α2)4b2a40.

    Case 2 corresponds the following solution for Kaup-Newell equation

    q(x,t)=(a0(b1+2b0)a0b0(1+Aeαxct)b0+b1(1+Aeαxct))×eι(aα2b202ba20xaα2(2b2a40+b40a2α2)4b2a40t). (3.9)

    Case 3.

    a0=b0a22b1,a1=a2(2b0b1)2b1,a2=a2,b0=b0,b1=b1.
    a=a,κ=2aα2b21ba22,ω=aα2(b2a42+8a2α2b41)2b2a42.

    Case 3 corresponds the following solution for Kaup-Newell equation

    q(x,t)=(b0a22b1+a2(2b0b1)2b1(1+Aeαxct)+a2(1+Aeαxct)2b0+b1(1+Aeαxct))×eι(2aα2b21ba22xaα2(b2a42+8a2α2b41)2b2a42t). (3.10)

    Case 4.

    a0=12a2,a1=a2,a2=a2,b0=1bb1,b1=b1.
    a=a,κ=2aα2b21ba22,ω=2aα2(b2a42+2a2α2b41)b2a42.

    Case 4 corresponds the following solution for Kaup-Newell equation

    q(x,t)=(12a2a2(1+Aeαxct)+a2(1+Aeαxct)21bb1+b1(1+Aeαxct))×eι(2aα2b21ba22x2aα2(b2a42+2a2α2b41)b2a42t). (3.11)

    The BAE with Kerr Law nonlinearity [56] is

    α1qxx+α2qxt+iqt+i(β1qxxx+β2qxxt)=i(λ(|q|2q)x+μ(|q|2)xq+θ|q|2qx). (4.1)

    Here q(x,t) representing the wave form. On the left of (4.1) α1 and α2 are the coefficients of GVD and STD, respectively. β1 and β2 are the coefficients of 3OD and STD, respectively. On the right of (4.1) μ and θ represents the outcome of nonlinear disperssion and λ represents the outcome of self-steepening in the nonappearance of SPM.

    Let us assumed that the solution of (4.1) is as under

    q(x,t)=g(η)eiϕ(x,t). (4.2)

    where

    ϕ(x,t)=κx+ωt+θ0,η=xvt. (4.3)

    Here g(η) shows amplitude, ϕ(x,t) is phase component. Also κ, v, θ0, ω denote the soliton frequency, speed, phase constant and wave number, respectively.

    Substituting (4.2) into (4.3) and splitting it into imaginary and real parts.

    The imaginary part has the form

    (β2vκ23β1κ2+2β2ωκv+2α2vκ2α1κ+α2ω)g+(β2v+β1)g(2μ+θ+3λ)g2g=0.

    We can get easily the value of v as under

    v=β1β2,

    and the constraints conditions as under

    {β2vκ2+2β2ωκ3β1κ2v2α1κ+2α2vκ+α2ω=0,3λ+2μ+θ=0.

    The real part has the form

    (α1α2v+3β1κ2β2vκωβ2)g(ω+α1κ2+β1κ3α2ωκβ2ωκ2)g(λ+θ)κg3=0. (4.4)

    Using balancing principal on (4.4), we attain M+1=N. So for M=1, we obtain N=2.

    Hence, solution of (4.4) by GKM as given in (2.4) will be reduced into the following form

    g(η)=a0+a1S(η)+a2S2(η)b0+b1S(η). (4.5)

    a0, a1, a2, b0 and b1 are constants.

    Substituting (4.5) into (4.4) and also applying (2.6), we acquire an expression in S(η). Collecting the coefficients of with same powerSi and equating to zero, the following system equations is got.

    {2ωβ2b21a24β2vκb21a2+κθa32+2α1b21a2+6β1κb21a22α2vb21a2+κλa32=0,3ωβ2b21a2+3α2vb21a2+6β2vκb21a29β1κb21a2+6α1b0a2b112β2vκb0a2b13α1b21a2+3κθa1a22+3κλa1a22+18β1κb0a2b16α2vb0a2b16ωβ2b0a2b1=0,α1b21a2+β2ωκ2b21a2+3κλa0a22α2vb21a2+α2ωκb21a2+18β1κa2b20+3κλa21a2+3κθa21a2+6α1a2b20+3κθa0a22β1κ3b21a26ωβ2a2b2027β1κb0a2b12β2vκb21a29α1b0a2b1+3β1κb21a26α2va2b2012β2vκa2b20+9α2vb0a2b1α1κ2b21a2ωβ2b21a2+18β2vκb0a2b1ωb21a2+9ωβ2b0a2b1=0,2β2vκb1a1b06β2vκb0a2b1+4β2vκb0b1a0+2α2ωκb0b1a2+2β2ωκ2b0b1a2+ωβ2b21a02ωβ2b20a12ωb0b1a2α1κ2b21a1β1κ3b21a1+α1b1a1b0+3α1b0a2b12α1b0b1a0+10α2va2b20+α2vb21a02α2vb20a130β1κa2b203β1κb21a0+6β1κb20a1+10ωβ2a2b2010α1a2b20ωb21a1+κθa31+κλa31+2α1b20a1α1b21a02α1κ2b0b1a22β1κ3b0b1a2+α2ωκb21a1+β2ωκ2b21a1α2vb1a1b03α2vb0a2b1+2α2vb0b1a0+3β1κb1a1b0+9β1κb0a2b16β1κb0b1a0+20β2vκa2b20+2β2vκb21a04β2vκb20a1ωβ2b1a1b03ωβ2b0a2b1+2ωβ2b0b1a0+6κλa0a1a2+6κθa0a1a2=0,2β2vκb1a1b06β2vκb0b1a0+2α2ωκb0b1a1+2β2ωκ2b0b1a1ωβ2b21a0+3ωβ2b20a1+3κλa20a2+3κλa0a21+3κθa20a2+3κθa0a212ωb0b1a1α1κ2b20a2α1κ2b21a0β1κ3b20a2β1κ3b21a0α1b1a1b0+3α1b0b1a04α2va2b20α2vb21a0+3α2vb20a1+12β1κa2b20+3β1κb21a09β1κb20a14ωβ2a2b20+4α1a2b20ωb21a0ωb20a23α1b20a1+α1b21a02β1κ3b0b1a1+α2ωκb20a2+α2ωκb21a0+β2ωκ2b20a2+β2ωκ2b21a0+α2vb1a1b03α2vb0b1a03β1κb1a1b0+9β1κb0b1a08β2vκa2b202β2vκb21a0+6β2vκb20a1+ωβ2b1a1b03ωβ2b0b1a02α1κ2b0b1a1=0,ωb20a1α1b0b1a0ωβ2b20a1α2vb20a1+ωβ2b0b1a0+α2ωκb20a1β1κ3b20a12β2vκb20a12β2vκb20a12α1κ2b0b1a0+α2vb0b1a0+3β1κb20a12β1κ3b0b1a02ωb0b1a0+α1b20a1α1κ2b20a1+3κθa20a13β1κb0b1a0+2β2vκb0b1a0+2α2ωκb0b1a0+β2ωκ2b20a1+2β2ωκ2b0b1a0+3κλa20a1=0,α2ωκb20a0+β2ωκ2b20a0+κθa30α1κ2b20a0β1κ3b20a0+κλa30ωb20a0=0. (4.6)

    On solving above system, get various types of solutions. These solutions are deliberated below.

    Case 1.

    κ=κ,λ=θ,
    ω=α2vωβ2+β2ωκ22β2vκβ1κ3+α2ωκ+3β1κ+α1α1κ2,
    a0=0,a1=a1,a2=0,b0=b1,b1=b1.

    Above these values correspond to the following solution for Biswas-Arshed equation.

    q(x,t)=a1b1[1Ae(xvt)]×ei{κx+ωt+θ0}. (4.7)

    Case 2.

    κ=κ,λ=12×2κθa20α1b203β1κb20+ωβ2b20+α2vb20+2β2vκb20κa20,
    ω=12α132β1κ+12ωβ2α1κ2+12α2vβ1κ3+β2ωκ2+β2vκ+α2ωκ,
    a0=a0,a1=a0(b1+2b0)b0,a2=0,b0=b0,b1=b1.

    Above these values correspond to the following solution for Biswas-Arshed equation

    q(x,t)=a0b0[b0Ae(xvt)(b0+b1)b0Ae(xvt)+(b0+b1)]×ei{κx+ωt+θ0}. (4.8)

    Case 3.

    κ=κ,λ=2α1b21κθa226β1κb21+2ωβ2b21+2α2vb21+4β2vκb21κa22,
    ω=α2ωκ+β2ωκ2α1κ2β1κ32α16β1κ+2ωβ2+2α2v+4β2vκ,
    a0=12a2,a1=a2,a2=a2,b0=12b1,b1=b1.

    Above these values correspond to the following solution for Biswas-Arshed equation

    q(x,t)=a2b1[1+A2e2(xvt)1A2e2(xvt)]×ei{κx+ωt+θ0}. (4.9)

    Case 4.

    κ=0,λ=λ,ω=α1ωβ2α2v,
    a0=a1,a1=a1,a2=0,b0=0,b1=b1.

    Above these values correspond to the following solution for Biswas-Arshed equation

    q(x,t)=a1b1[Ae(xvt)]×ei{ωt+θ0}. (4.10)

    Case 5.

    κ=α1+ωβ2+α2v3β1+2β2v,λ=θ,

    ω=1(3β1+2β2v)3×[4α21α2v2β2+2α1α22v3β29α2ωα1β21+9α2ω2β2β21+9α22ωvβ21+2α22ωv3β22+6ωβ2α21β16ωβ22α21v6ω2β22α1β1+6ω2β32α1v+3β1α21α2v+2ω3β32β12ω3β42vβ1α32v3+2α31β2v+4α2ωα1β22v29α2ω2β22β1v12α22ωv2β1β22β1α31+6α2α1β1β2v],

    a0=0,a1=a1,a2=a2,b0=b0,b1=b1.

    Above these values correspond to the following solution for Biswas-Arshed equation

    q(x,t)=[a1Ae(xvt)+(a1+a2)A2e2(xvt)+(2b0+b1)Ae(xvt)+(b0+b1)]×ei{κx+ωt+θ0}. (4.11)

    In this study, we effectively construct novel exact solutions in form of optical solitons for Kaup-Newell equation and Biswas-Arshed equation using the generalized Kudryashov method. This method is considered as most recent scheme in this arena and that is not utilized to this equation earlier. For physical analysis, 3-dim, 2-dim and contour plots of some of these solutions are included with appropriate parameters. These acquired solutions discover their application in communication to convey information because solitons have the capability to spread over long distances without reduction and without changing their forms. Acquired results are novel and distinct from that reported results. In this paper, we only added particular figures to avoid overfilling the document. For graphical representation for KNE and BAE, the physical behavior of (3.8) using the proper values of parameters α=0.3, a1=0.65, b1=0.85, p=0.98, q=0.95, k=2, A=3, b=2, c=4. and t=1 are shown in Figure 1, the physical behavior of (3.9) using the appropriate values of parameters α=0.75, a0=1.5, b0=1.7, b1=0.98, A=3, b=1.6, a0=2, c=2.5. and t=1 are shown in Figure 2, the physical behavior of (3.11) using the proper values of parameters α=0.75, a0=1.5, b0=1.7, b1=0.98, A=3, b=1.6, a0=2, c=2.5. and t=1 are shown in Figure 3, the absolute behavior of (4.9) using the proper values of parameters α=0.75, a0=1.5, b0=1.7, A=2.3, b=1.6, c=2.5, v=2.5, θ0=4. and t=1 are shown in Figure 4.

    Figure 1.  (A): 3D graph of (3.8) with α=0.3, a1=0.65, b1=0.85, p=0.98, q=0.95, k=2, A=3, b=2, c=4. (A-1): 2D plot of (3.8) with t=1. (A-2): Contour graph of (3.8).
    Figure 2.  (B): 3D graph of (3.9) with α=0.75, a0=1.5, b0=1.7, b1=0.98, A=3, b=1.6, a0=2, c=2.5. (B-1): 2D plot of (3.9) with t=1. (B-2): Contour graph of (3.9).
    Figure 3.  (C): 3D graph of (3.11) with α=0.75, a0=1.5, b0=1.7, b1=0.98, A=3, b=1.6, a0=2, c=2.5. (C-1): 2D plot of (3.11) with t=1. (C-2): Contour graph of (3.11).
    Figure 4.  (D): 3D graph of (4.9) with α=0.75, a0=1.5, b0=1.7, A=2.3, b=1.6, c=2.5, v=2.5, θ0=4. (D-1): 2D plot of (4.9) with t=1. (D-2): Contour graph of (4.9).

    The study of the exact solutions of nonlinear models plays an indispensable role in the analysis of nonlinear phenomena. In this work, we have constructed and analyzed the optical solitons solutions of the Kaup-Newell equation and Biswas-Arshad equation by using Kudryashove method. The transmission of ultrashort optical solitons in optical fiber is modeled by these equations. We have achieved more general and novel exact solutions in the form of dark, singular and bright solitons. The obtained solutions of this article are very helpful in governing solitons dynamics. The constructed solitons solutions approve the effectiveness, easiness and influence of the under study techniques. we plotted some selected solutions by giving appropriate values to the involved parameters. The motivation and purpose of this study is to offer analytical techniques to discover solitons solutions which helps mathematicians, physicians and engineers to recognize the physical phenomena of these models. This powerful technique can be employed for several other nonlinear complex PDEs that are arising in mathematical physics. Next, the DNLSE classes II and III will be scrutinized via the similar methods to more evaluate them, this definitely will offer a huge understanding of the methods along with the classes of DNLSE. These solutions may be suitable for understanding the procedure of the nonlinear physical phenomena in wave propagation.

    The authors would like to express their sincere thanks to the Department of Mathematics, University of Management and Technology Lahore, Pakistan.

    The authors declare that they have no conflict of interest.



    [1] S. Zhang, T. C. Xia, A generalized new auxiliary equation method and its applications to nonlinear partial differential equations, Phys. Lett. A, 363 (2007), 356–360. doi: 10.1016/j.physleta.2006.11.035
    [2] Sirendaoreji, S. Jiong, Auxiliary equation method for solving nonlinear partial differential equations, Phys. Lett. A, 309 (2003), 387–396. doi: 10.1016/S0375-9601(03)00196-8
    [3] M. L. Wang, Y. B. Zhou, Z. B. Li, Application of a homogeneous balance method to exact solutions of nonlinear equations in mathematical physics, Phys. Lett. A, 216 (1996), 67–75. doi: 10.1016/0375-9601(96)00283-6
    [4] M. L. Wang, Exact solutions for a compound KdV-Burgers equation, Phys. Lett. A, 213 (1996), 279–287. doi: 10.1016/0375-9601(96)00103-X
    [5] X. H. Wu, J. H. He, EXP-function method and its application to nonlinear equations, Chaos Solitons Fract., 38 (2008), 903–910. doi: 10.1016/j.chaos.2007.01.024
    [6] X. H. Wu, J. H. He, Solitary solutions, periodic solutions and compacton-like solutions using the Exp-function method, Comput. Math. Appl., 54 (2007), 966–986. doi: 10.1016/j.camwa.2006.12.041
    [7] H. A. Abdusalam, On an improved complex tanh-function method, Int. J. Nonlin. Sci. Num., 6 (2005), 99–106.
    [8] H. C. Hu, X. Y. Tang, S. Y. Lou, Q. P. Liu, Variable separation solutions obtained from Darboux transformations for the asymmetric Nizhnik-Novikov-Veselov system. Chaos Solitons Fract., 22 (2004), 327–334.
    [9] S. B. Leble, N. V. Ustinov, Darboux transforms, deep reductions and solitons, J. Phys. A: Math. Gen, 26 (1993), 5007–5016. doi: 10.1088/0305-4470/26/19/029
    [10] J. Lee, R. Sakthivel, New exact travelling wave solutions of bidirectional wave equations, Pramana-J. Phys., 76 (2011), 819–829. doi: 10.1007/s12043-011-0105-4
    [11] A. Bekir, O. Unsal, Analytic treatment of nonlinear evolution equations using first integral method, Pramana-J. phys., 79 (2012), 3–17. doi: 10.1007/s12043-012-0282-9
    [12] S. Abbasbandy, A. Shirzadi, The first integral method for modified Benjamin-Bona-Mahony equation, Commun. Nonlinear Sci., 15 (2010), 1759–1764. doi: 10.1016/j.cnsns.2009.08.003
    [13] E. J. Parkes, B. R. Duffy, P. C. Abbott, The Jacobi elliptic-function method for finding periodic-wave solutions to nonlinear evolution equations, Phys. Lett. A, 295 (2002), 280–286. doi: 10.1016/S0375-9601(02)00180-9
    [14] S. K. Liu, Z. T. Fu, S. D. Liu, Q. Zhao, Jacobi elliptic function expansion method and periodic wave solutions of nonlinear wave equations, Phys. Lett. A, 289 (2001), 69–74. doi: 10.1016/S0375-9601(01)00580-1
    [15] A. J. M. Jawad, M. D. Petkovic, A. Biswas, Modified simple equation method for nonlinear evolution equations, Appl. Math. Comput., 217 (2010), 869–877.
    [16] K. Khan, M. A. Akbar, Exact and solitary wave solutions for the Tzitzeica-Dodd-Bullough and the modified KdV-Zakharov-Kuznetsov equations using the modified simple equation method, Ain Shams Eng. J., 4 (2013), 903–909. doi: 10.1016/j.asej.2013.01.010
    [17] K. Khan, M. A. Akbar, Exact solutions of the (2+1)-dimensional cubic Klein-Gordon equation and the (3+1)-dimensional Zakharov-Kuznetsov equation Using the modified simple equation method, J. Assoc. Arab Uni. Basic Appl. Sci., 15 (2014), 74–81.
    [18] K. Khan, M. A. Akbar, Application of exp(-φ(ξ))-expansion method to find the exact solutions of modified Benjamin-Bona- Mahony equation, World Appl. Sci. J., 24 (2013), 1373–1377.
    [19] M. L. Wang, X. Z. Li, J. L. Zhang, The GG-expansion method and travelling wave solutions of nonlinear evolution equations in mathematical physics, Phys. Lett. A, 372 (2008), 417–423. doi: 10.1016/j.physleta.2007.07.051
    [20] H. Kim, R. Sakthivel, New exact travelling wave solutions of some nonlinear higher dimensional physical models, Rep. Math. Phys., 70 (2012), 39–50. doi: 10.1016/S0034-4877(13)60012-9
    [21] K. Khan, M. A. Akbar, Traveling wave solutions of nonlinear evolution equations via the enhanced (G/G)-expansion method, J. Egypt. Math. Soc., 22 (2014), 220–226. doi: 10.1016/j.joems.2013.07.009
    [22] M. E. Islam, K. Khan, M. A. Akbar, R. Islam, Traveling wave solutions of nonlinear evolution equations via enhanced (G/G)- expansion method, GANIT J. Bangladesh Math. Soc., 33, (2013), 83–92.
    [23] H. Naher, F. A. Abdullah, New generalized and improved (G/G)-expansion method for nonlinear evolution equations in mathematical physics, J. Egypt. Math. Soc., 22 (2014), 390–395. doi: 10.1016/j.joems.2013.11.008
    [24] H. Naher, F. A. Abdullah, A. Bekir, Abundant traveling wave solutions of the compound KdV-Burgers equation via the improved (G/G)-expansion method, AIP Adv., 2 (2012), 042163. doi: 10.1063/1.4769751
    [25] H. Naher, F. A. Abdullah, Some new traveling solutions of the nonlinear reaction diffusion equation by using the improved (G/G)-expansion method, Math. Probl. Eng., 2012 (2012), 871724.
    [26] H. Naher, F. A. Abdullah, M. A. Akbar, Generalized and improved (G/G)-expansion method for (3+ 1)-dimensional modified KdV-Zakharov-Kuznetsev equation, PloS One, 8 (2013), e64618. doi: 10.1371/journal.pone.0064618
    [27] Y. Molliq R, M. S. M. Noorani, I. Hashim, Variational iteration method for fractional heat-and wave-like equations, Nonlinear Anal-Real., 10 (2009), 1854–1869. doi: 10.1016/j.nonrwa.2008.02.026
    [28] S. T. Mohiud-Din, M. A. Noor, Homotopy perturbation method for solving fourth-order boundary value problems, Math. Probl. Eng., 2007 (2007), 098602.
    [29] S. T. Mohyud-Din, M. A. Noor, Homotopy perturbation method for solving partial differential equations, Z. Naturforsch, 64a (2009), 157–170.
    [30] S. T. Mohyud-Din, A. Yildrim, S. Sariaydin, Approximate series solutions of the viscous Cahn-Hilliard equation via the homotopy perturbation method, World Appl. Sci. J., 11 (2010), 813–818.
    [31] C. Changbum, R. Sakthivel, Homotopy perturbation technique for solving two-point boundary value problems-comparison with other methods, Comput. Phys. Commun., 181 (2010), 1021–1024. doi: 10.1016/j.cpc.2010.02.007
    [32] R. Sakthivel, C. Changbum, L. Jonu, New travelling wave solutions of Burgers equation with finite transport memory, Z. Naturforsch, 65 (2010), 633–640. doi: 10.1515/zna-2010-8-903
    [33] Y. M. Zhao, F-expansion method and its application for finding new exact solutions to the Kudryashov-Sinelshchikov equation, J. Appl. Math., 2013 (2013), 895760.
    [34] H. W. Hua, A generalized extended F-expansion method and its application in (2+1)-dimensional dispersive long wave equation, Commun. Theor. Phys., 46 (2006), 580. doi: 10.1088/0253-6102/46/4/002
    [35] M. S. Islam, k. Khan, M. A. Akbar, A. Mastroberardino, A note on improved F-expansion method combined with Riccati equation applied to nonlinear evolution equations, R. Soc. Open Sci., 1, (2014), 140038.
    [36] W. J. Liu, Y. N. Zhu, M. L. Liu, B. Wen, S. B. Fang, H. Teng, et al., Optical properties and applications for MoS2-Sb2Te3-MoS2 heterostructure materials, Photonics Res., 6 (2018), 220–227. doi: 10.1364/PRJ.6.000220
    [37] W. J. Liu, L. H. Pang, H. N. Han, M. L. Liu, M. Lei, S. B. Fang, et al., Tungsten disulfide saturable absorbers for 67 fs mode-locked erbium-doped fiber lasers, Opt. Express, 25 (2017), 2950–2959. doi: 10.1364/OE.25.002950
    [38] N. Raza, S. Arshed, A. Javid, Optical solitons and stability analysis for the generalized second-order nonlinear Schrodinger equation in an optical fiber, Int. J. Nonlin. Sci. Num., 21 (2020), 855–863. doi: 10.1515/ijnsns-2019-0287
    [39] W. J. Liu, W. T. Yu, C. Y. Yang, M. L. Liu, Y. J. Zhang, M. Lei, Analytic solutions for the generalized complex Ginzburg-Landau equation in fiber lasers, Nonlinear Dyn., 89 (2017), 2933–2939. doi: 10.1007/s11071-017-3636-5
    [40] X. Y. Fan, T. Q. Qu, S. C. Huang, X. X. Chen, M. H. Cao, Q. Zhou, et al., Analytic study on the influences of higher-order effects on optical solitons in fiber laser, Optik, 186 (2019), 326–331. doi: 10.1016/j.ijleo.2019.04.102
    [41] Y. Khan, Fractal modification of complex Ginzburg-Landau model arising in the oscillating phenomena, Results Phys., 18 (2020), 103324. doi: 10.1016/j.rinp.2020.103324
    [42] M. Arshad, A. R. Seadawy, D. C. Lu, Exact bright-dark solitary wave solutions of the higher-order cubic-quintic nonlinear Schrodinger equation and its stability, Optik, 138 (2017), 40–49. doi: 10.1016/j.ijleo.2017.03.005
    [43] A. R. Seadawy, M. Arshad, D. C. Lu, Modulation stability analysis and solitary wave solutions of nonlinear higher-order Schrodinger dynamical equation with second-order spatiotemporal dispersion, Indian. J. Phys., 93 (2019), 1041–1049. doi: 10.1007/s12648-018-01361-y
    [44] K. Porsezian, A. K. Shafeeque Ali, A. I. Maimistov, Modulation instability in two-dimensional waveguide arrays with alternating signs of refractive index, J. Opt. Soc. Am. B, 35 (2018), 2057–2064. doi: 10.1364/JOSAB.35.002057
    [45] A. Hasegawa, F. Tappert, Transmission of stationary nonlinear optical pulses in dispersive dielectric fibers, Appl. Phys. Lett., 23 (1973), 142. doi: 10.1063/1.1654836
    [46] A. Ankiewicz, D. J. Kedziora, A. Chowdury, U. Bandelow, N. Akhmediev, Infinite hierarchy of nonlinear Schrodinger equations and their solutions, Phys. Rev. E, 93 (2016), 012206. doi: 10.1103/PhysRevE.93.012206
    [47] A. Ankiewicz, N. Akhmediev, Higher-order integrable evolution equation and its soliton solutions, Phys. Lett. A, 378 (2014), 358–361. doi: 10.1016/j.physleta.2013.11.031
    [48] M. A. Banaja, S. A. Alkhateeb, A. A. Alshaery, E. M. Hilal, A. H. Bhrawy, L. Moraru, et al., Optical solitons in dual-core couplers, Wulfenia J., 21 (2014), 366–380.
    [49] A. H. Bhrawy, A. A. Alshaery, E. M. Hilal, M. Savescu, D. Milovic, K. R. Khang, et al., Optical solitons in birefringent fibers with spatio-temporal dispersion, Optik, 125 (2014), 4935–4944. doi: 10.1016/j.ijleo.2014.04.025
    [50] T. A. Sulaiman, T. Akturk, H. Bulut, H. M. Baskonus, Investigation of various soliton solutions to the Heisenberg ferromagnetic spin chain equation, J. Electromagnet Wave., 32 (2017), 1093–1105.
    [51] M. A. Banaja, A. A. Al Qarni, H. O. Bakodah, Q. Zhou, S. P. Moshokoa, A. Biswas, The investigate of optical solitons in cascaded system by improved adomian decomposition scheme, Optik, 130 (2017), 1107–1114. doi: 10.1016/j.ijleo.2016.11.125
    [52] A. R. Seadawy, D. C. Lu, Bright and dark solitary wave soliton solutions for the generalized higher order nonlinear Schrodinger equation and its stability, Results Phys., 7 (2017), 43–48. doi: 10.1016/j.rinp.2016.11.038
    [53] A. Biswas, S. Arshed, Optical solitons in presence of higher order dispersions and absence of self-phase modulation, Optik, 174 (2018), 452–459. doi: 10.1016/j.ijleo.2018.08.037
    [54] M. Kaplan, A. Bekir, A. Akbulut, A generalized Kudryashov method to some nonlinear evolution equations in mathematical physics, Nonlinear Dyn., 85 (2016), 2843–2850. doi: 10.1007/s11071-016-2867-1
    [55] A. Biswas, M. Ekici, A. Sonmezoglu, R. T. Alqahtanib, Sub-pico-second chirped optical solitons in mono-mode fibers with Kaup-Newell equation by extended trial function method, Optik, 168 (2018), 208–216. doi: 10.1016/j.ijleo.2018.04.069
    [56] H. Ur Rehman, N. Ullah, M. A. Imran, Optical solitons of Biswas-Arshed equation in birefringent fibers using extended direct algebraic method, Optik, 226 (2021), 165378. doi: 10.1016/j.ijleo.2020.165378
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