Bifurcation Analysis and Bounded Optical Soliton Solutions of the Biswas-Arshed Model

1  Introduction

The study of optical solitons and their applications in the account of optical fiber transmissions is a paramount topic in communication networks. The general concepts of transmission optical solitons in nonlinear optical fiber systems are fundamentally important in controlling optical continuum creation and transferring information over very long distances. The general nonlinear complex models (see [1–7]) are the best examples of the exploration and description of the effect on the picosecond vibration with the group velocity dispersion (GVD) as well as the self-phase modulation (SPM). Such models can be used to explicitly address the electro-magnetic waves with ultrashort pules that can cover the world within a nanosecond. Thus these varieties of waves have been highly studied in the communication technologies, especially, in the optical fibers, data transmissions, telecommunication sectors, transoceanic spaces and so on [6–10]. Recently, Biswas and Arshed introduced a model which is greatly interesting due to the consideration of minor GVD and neglecting the SPM [1]. A number of works have done on this model to investigate optical solitons by using various approaches such as the trail solution technique [12], modified simple equation technique [13], Kerr and power law nonlinearity [14,15], mapping method [16], the extended trial function method [17], parameter restriction approach [18] and the tan(θ2) expansion approach [19]. They also pointed out the bright, singular and combo-solitons for the two integration structures of the model. It is fundamentally effective in investigating the dynamical bounded and unbounded solitons in optical fibers and metameterials in case of both low GVD and nonlinearity via new effective approaches. Young dynamical researchers established more new effective schemes to obtain various types of optical solitons like the generalized Kudryashov (GK) approaches [20,21], the Jacobian elliptic function method [22], the extended sinh-Gordon equation expansion method [23], Hirota bilinear scheme [24,25], Darboux transformation method [26], dynamical system approaces [27], solitary waves travel in a porous medium or along an unsmooth boundary [28,29] and so on. Although all those approaches have derived many profound results, some bounded optical wave solutions could still be an unexplored subjects via dynamical scheme with a bifurcation of the BAM.

The purpose of this paper is to show how we can present optical shock wave including both bright and dark, optical soliton solutions, and construct the periodic wave solutions via dynamical system method with bifurcation analysis for the model. We also use the extended sinh-Gordon equation expansion (EShGEE) and the generalized Kudryashov (GK) approaches [20,21] to get a more bounded wave solution. To our knowledge, these types of investigations for the BAM are the first step in the study of the dynamical system, EShGEE and the GK methods.

2  The Biswas-Arshed Model

The Biswas-Arshed model (BAM) [11] is given as the following form:

iΨt+a1Ψxx+a2Ψxt+i(b1Ψxxx+b2Ψxxt)=i[ε(|Ψ|2Ψ)x+σ(|Ψ|2)xΨ+ϑ|Ψ|2Ψx],(1)

where Ψ(x,t) is the wave function of soliton and a1, a2, b1, b2 are, respectively, the GVD coefficient, the spatio-temporal dispersion, the third-order dispersion coefficient, and the spatio-temporal third-order dispersion coefficient. In the model ε represents self-steepening and σ,ϑ are dispersion effects. Biswas and Arshed first explored this model in the context of higher-order dispersions with minor self-phase modulation. The optical solutions of the model with transmission wave have been retrieved in the point of view of Kerr and non-Kerr law [14]. Recently, the effect of the optical shock wave, optical solitons, rogue waves and their various interactions were investigated in [19]. In this paper, we revisit this model as it has wider applicability, and many other aspects such as bright and dark, optical soliton and the periodic wave solutions of the BAM by the method of a dynamical system with bifurcation analysis, the EShGEE and the GK methods still remain. In the following section, we use the transformation variable to Eq. (1).

2.1 The Structure of BAM

The transformation variable [11]

Ψ(x,t)=Φ(ζ)eiχ(x,t),(2)

where Φ(ζ) is the amplitude portion with ζ=x−δt, the phase component χ(x,t)=−kx+wt+ρ, where the constants δ,ρ,w,k are, respectively, the soliton velocity, the phase constant, the wave number and the frequency of the soliton, convert the nonlinear evaluation Eq. (1) to an ordinary differential equation. After a long computation and an integration, one can present the differential equation of Eq. (1) [11,14] as follows:

(a1−a2δ+3b1k−2b2δk−b2w)Φ′′−(a1k2−a2wk+b1k3−b2wk2+w)Φ−k(ε+ϑ)Φ3=0.(3)

and the imaginary part:

(b2δk2−3b1k2+2b2wk−δ+2a2δk−2a1k+a2w)Φ′+(b1−b2δ)Φ′′′−(2σ+ϑ+3ε)Φ2Φ′=0.(4)

One can easily find the following conditions that satisfies the Eq. (4): δ=b1/b2,b2δk2−3b1k2+2b2wk−δ+2a2δk−2a1k+a2w=0 and 2σ+ϑ+3ε=0.

Thus, we have to analyze the Eq. (3) only. In the following subsection, we analyze bifurcation to acquire phase portraits to identify the number and types of solutions that exist for the model. Beside this, we apply three different techniques, namely, the dynamical, the EShGEE and the GK approaches to the corresponding model Eq. (3) in order to obtain the corresponding exact optical soliton solutions of Eq. (1).

2.2 Bifurcations Analysis of the BAM

We first consider the bifurcations of phase orbits of ordinary differential equation (ODE) Eq. (3). To proceed of the motive, we need to convert the second order ODE to a dynamical system, which is possible by setting Φ′=Θ in the ODE. In this case the ODE in Eq. (3) takes the form of a dynamical system:

Φ′=Θ,Θ′=QPΦ+RPΦ3,(5)

with the Hamiltonian

H(Φ,Θ)=Θ22−Q2PΦ2−R4PΦ4,(6)

where P=a1−a2δ+3b1k−2b2δk−b2w, Q=a1k2−a2wk+b1k3−b2wk2+w and R=k(ε+ϑ).

Setting the system of Eq. (5) to zero gives critical points O(0,0), A(−QR,0) and B(−−QR,0). Determinants of Jacobian matrices at the three equilibrium points are: detJO=−QP, detJA=2QP and detJB=2QP.

If QR<0, the system has three equilibrium points O,A,B and two cases arise here: (i) QP>0 implies O-saddle, A,B-centers; there exist two homoclinic orbits ΓA (right side) and ΓB (left side) (see the Fig. 1a) that connect at the saddle O. Centers A and B are surrounded by a family of periodic orbits

ΓA(h)={H(Φ,Θ)=h,h∈(Q22RP,0)}ΓB(h)={H(Φ,Θ)=h,h∈(Q22RP,0)}.(7)

Figure 1: Bifurcation of system Eq. (5) with QR<0: (a) for QP>0 (b) for QP<0 at x=t=1

(ii) QP<0 implies O-center, A, B-saddles. There exist two heteroclinic orbits ΓuO (upper) and ΓLO (lower) (see the Fig. 1b) that connect at the saddles A, B. Center O is surrounded by a family of periodic orbits.

ΓO(h)={H(Φ,Θ)=h,h∈(0,Q22RP)}(8)

On the other hand, if QR>0, the system has only one real equilibrium point at O: (i) for QP>0 implies O-saddle (see the Fig. 2a), (ii) for QP<0 implies O-center (see the Fig. 2b).

Figure 2: Bifurcation of system Eq. (5) with QR>0 : (a) for QP>0 (b) for QP<0 at x=t=1

Moreover, a different dynamical system can arise for Q=0, which reads

Φ′=Θ,Θ′=RPΦ3,(9)

with the Hamiltonian

H(Φ,Θ)=Θ22−R4PΦ4.(10)

It has a unique critical point A(0,0) of higher order with detP=0. From the theorem-2 of [27], we get a2m+1=2RP and bn=0. It yields that A is a saddle point when RP>0 (see the Fig. 3a) and A is a critical point when RP<0 (see the Fig. 3b). We observe that all bounded orbits and traveling wave solutions can be simulated on the phase portraits. All bounded orbits and travelling wave solutions can be simulated.

Figure 3: Bifurcation of system Eq. (5) with Q=0: (a) for QP>0 (b) for QP<0 at x=t=1

2.3 Bounded Travelling Wave Solutions of BAM

This section will provide the explicit expressions of all bounded travelling wave solution of the system Eq. (5).

2.3.1 The Periodic Wave Solution

Recall the cases QR<0 and QP>0, there are two classes of periodic orbits ΓA and ΓB enclosing A and B, respectively. The corresponding wave solutions via Eq. (6) closed orbit ΓA are as follows:

Θ=±−R2P(Φ−l)(Φ−m)(Φ−n)(r−Φ),(11)

where l<m<n<Φ<r. Let us consider the periodicity of the orbit is 2T1 and initial value Φ(0)=n, we have

∫nΦ−2PRdΦ(Φ−l)(Φ−m)(Φ−n)(r−Φ)=∫0ζdζ;0<ζ<T1.(12)

−∫Φn−2PRdΦ(Φ−l)(Φ−m)(Φ−n)(r−Φ)=∫ζ0dζ;−T1<ζ<0.(13)

Combining Eqs. (12) and (13), it leads to

∫nΦ−2PRdΦ(Φ−l)(Φ−m)(Φ−n)(r−Φ)=|ζ|.(14)

By the direct elliptic integral and calculation, we arrive at the solution

Φ1(ζ)=m+(n−m)(r−m){(r−m)−(r−n)sn2(−R2P(r−m)(n−l)2|ζ|)}−1,(15)

where T1=2b2πb1((r−m)(n−l))−2PR. Noting that

∫nΦdΦ(Φ−l)(Φ−m)(Φ−n)(r−Φ)=κsn−1((r−m)(Φ−n)(r−n)(Φ−m),β),(16)

where κ=2/(r−m)(n−l) and β2=(r−n)(m−l)/((r−m)(n−l)). Now, the corresponding bounded periodic optical wave solution is coming from the relation Eq. (2) with modulus |Ψ1(x,t)|.

Similarly, the periodic solution corresponding to ΓB can be expressed as

Θ=±−R2P(Φ−l)(m−Φ)(n−Φ)(r−Φ),(17)

where l<Φ<m<n<r. Taking period is 2T2 and initial condition Φ(0)=l, we arrive at the solution

Φ2(ζ)=r−(r−m)(r−l){(r−m)+(m−l)sn2(−R2P(r−m)(r−l)2|ζ|)}−1,(18)

where κ=2/(r−m)(n−l) and β2=(r−n)(m−l)/((r−m)(n−l)), and T2=2b2πb1((r−m)(r−l))−2PR. Now, the corresponding bounded periodic wave are coming from the relation Eq. (2) with modulus |Ψ2(x,t)|.

Again, recall the cases dfracQR<0 and QP<0, there is class of periodic orbits ΓO enclosing O. The corresponding wave solutions via Eq. (6) closed orbit ΓO are as follows:

Θ=±R2P(Φ−l)(Φ−m)(n−Φ)(r−Φ),(19)

where l<m<Φ<n<r which are real valued constants. Let us consider the periodicity of the orbit is 2T3 and initial value Φ(0)=m, we obtain

∫mΦ2PRdΦ(Φ−l)(Φ−m)(n−Φ)(r−Φ)=∫0ζdζ;0<ζ<T3.(20)

−∫Φm2PRdΦ(Φ−l)(Φ−m)(n−Φ)(r−Φ)=∫ζ0dζ;−T3<ζ<0.(21)

Combining Eqs. (20) and (21), it leads to

∫mΦ2PRdΦ(Φ−l)(Φ−m)(n−Φ)(r−Φ)=|ζ|.(22)

Again by the elliptic integral and after some calculation, we arrive at the solution

Φ3(ζ)=l+(n−l)(m−l){(n−l)−(n−m)sn2(R2P(r−m)(n−l)2|ζ|)}−1,(23)

where T3=2b2πb1((r−m)(n−l))2PR. Noting that

∫mΦdΦ(Φ−l)(Φ−m)(n−Φ)(r−Φ)=κsn−1((n−l)(Φ−m)(n−m)(Φ−l),β),(24)

where κ=2/(r−m)(n−l) and β2=(n−m)(r−l)/((r−m)(n−l)). Now, corresponding bounded periodic wave are coming from the relation Eq. (2) with modulus |Ψ3(x,t)|. It is noted that three classes of periodic wave solutions of the Eq. (5) specified by Φi(x,t);i=1,2,3 are achieved for different regions and the corresponding optical waves, which are specified by |Ψi(x,t)|,i=1,2,3. Since the nature of the waves are similar but in different boundary area with different initial conditions, we depicted only the Φ1(x,t) and Ψ1(x,t) graphically in the Fig. 4. To investigate all the propagation properties of the wave, we illustrate here the 3D plots of Φ1(x,t), real part, imaginary part and square of modulus of |Ψ1(x,t)| in Figs. 4a–4d, respectively.

Figure 4: (a) Periodic wave for + via Eq. (15), (b) real part of Ψ1(x,t) for Eq. (15), (c) imaginary part of Ψ1(x,t) for Eq. (15) and (d) optical bright waves |Ψ1(x,t)|2 for the parametric values a1=5,a2=b2=2,b1=δ=k=ε=1,w=0.5,ϑ=−2,l=0.2,m=0.3,n=0.4,r=3

2.3.2 The Solitary Wave Solutions

Using the cases QR<0 and QP>0, the homoclinic orbits ΓA and ΓB can be represented as

Θ=±−R2P(Φ+l)Φ2(l−Φ),(25)

where −l<Φ<l. Setting the homoclinic orbit and initial value Φ(0)=m, we obtain

∫Φl−2PRdΦΦ(Φ+l)(l−Φ)=−|ζ|.(26)

Applying the elliptic integral and some calculation, we arrive at the solution

Φ4(ζ)=2lexp(−R2Pm|ζ|)1+exp(2−R2Pm|ζ|),l>0.(27)

The resulting solution Φ4 comes due to the homoclinic orbit ΓA for the system Eq. (5) by (6) and it expresses the bright peaked soliton, i.e., bright peakon. Now, the corresponding bounded optical soliton solution will come through the relation Eq. (2) with modulus as |Ψ4(x,t)|, which is also a bright optical peakon soliton.

Similarly, let the homoclinic orbit and initial value Φ(0)=−m, we obtain

∫−lΦ−2PRdΦΦ(Φ+l)(l−Φ)=|ζ|.(28)

Through the elliptic integral and after some calculation, we arrive at the solution

Φ5(ζ)=−2le−R2Pm|ζ|1+e2−R2Pm|ζ|,l>0.(29)

The obtained solution Φ5 comes due to the homoclinic orbit ΓB for the system Eq. (5) by Eq. (6) and it expresses the dark peaked soliton, i.e., anti-peakon. Now, the corresponding bounded optical soliton solution will arise through the relation Eq. (2) with modulus as |Ψ5(x,t)|, which is also a bright optical peakon soliton. The nature of the peaked solitons and its optical peaked solitons are demonstrated via the Figs. 5 and 6 for Φ4(x,t),Ψ4(x,t) and Φ4(x,t),Ψ4(x,t) respectively. To investigate the all propagation properties of the wave, we illustrate here the 3D plots of Φ4(x,t),Φ5(x,t), real part, imaginary part and square of modulus of |Ψ4(x,t)|,|Ψ5(x,t)| in Figs. 5 and 6a–6d, respectively.

Figure 5: (a) Bright peaked soliton via Eq. (27), (b) Real part of Ψ4(x,t) for Eq. (27), (c) Imaginary part of Ψ4(x,t) for Eq. (27) and (d) Optical bright peaked soliton |Ψ4(x,t)|2 for the parametric values a1=5,a2=b2=2,b1=δ=k=ε=1,w=0.5,ϑ=−2,l=0.2,m=5

Figure 6: (a) Dark peaked soliton via Eq. (29), (b) Real part of Ψ4(x,t) for Eq. (29), (c) Imaginary part of Ψ4(x,t) for Eq. (29) and (d) Optical bright peaked soliton |Ψ4(x,t)|2 for the parametric values a1=5,a2=b2=2,b1=δ=k=ε=1,w=0.5,ϑ=−2,l=0.2,m=5

2.3.3 The Shock Wave Solutions

For the cases QR<0 and QP<0, the heteroclinic orbits Γμ and ΓL can be represented as

Θ=±R2P(Φ−l)2(r−Φ)2,(30)

where −−QR=l<Φ<r=−QR. For the heteroclinic orbit and initial value Φ(0)=l+r2=0, we obtain

∫0Φ2PRdΦ(Φ−l)(r−Φ)=∫0ζdζ;−∞<ζ<+∞.(31)

−∫0Φ2PRdΦ(Φ−l)(r−Φ)=∫0ζdζ;−∞<ζ<+∞,(32)

and applying the elliptic integral and after some calculation, we arrive at the solution

Φ6(ζ)=r−l2tanh(r−l2R2Pζ),Φ7(ζ)=r−l2tanh(l−r2R2Pζ)(33)

Noting that

∫0ΦdΦ(Φ−l)(Φ−r)=2r−ltanh−1(2Φ−(l+r)r−l).(34)

The resulting solution Φ6(x,t) comes in-terms of tanh-function due to the heteroclinic orbit Γμ for the system Eq. (5) by Eq. (6) and it expresses the kink type shock wave, and its corresponding bounded optical soliton solution will arise through the relation Eq. (2) with modulus as |Ψ6(x,t)|, which is a bounded optical dark peaked soliton, i.e., anti-peakon. The nature of this soliton is specified in Figs. 7a and 7b for Φ6(x,t) and |Ψ6(x,t)|2, respectively. Moreover, the resulting solution Φ7(x,t) (depicted in 7c) comes in-terms of tanh-function due to the heteroclinic orbit ΓL for the system Eq. (5) by Eq. (6), and it expresses the anti-kink type shock wave and its corresponding bounded optical soliton solution comes through the relation Eq. (2) with modulus as |Ψ7(x,t)|2 (depicted in Fig. 7d), which is also a bounded optical dark peaked soliton, i.e., anti-peakon.

Figure 7: (a) Shock wave (kink) via Φ(x,t) for Eq. (33), (b) Optical anti-peakon |Ψ6(x,t)|2 for Eq. (29), (c) Shock wave (anti-kink) via Φ(x,t) for Eq. (29) and (d) Optical anti-peakon |Ψ7(x,t)|2 for the parametric values a1=b1=δ=k=ε=1,a2=b2=2,w=0.5,ϑ=−2

2.4 Optical Soliton Solutions to BAM via the EShGEE

One considers the general form of the trail solution in the extended sinh-Gordon equation expansion approach (EShGEE) [23],

Ψ(⊖)=∑r=1ncoshr−1(⊖)[Lrsinh⁡(⊖)+Mrcosh⁡(⊖)]+M0,(35)

where M0, Mr, Lr, r=1,2,…,n are free constants to be later calculated, and the ⊖ is a function of ζ, which satisfies the condition,

d⊖dζ=sinh⁡(⊖).(36)

Due to balance principal, the value n of Eq. (35) can be obtained. The Eq. (36) has been obtained from the sinh-Gordon equation [23],

uxt=λsinh⁡(u),(37)

and they [23] obtained the solutions,

sinh⁡(⊖)=±csch(ζ),orsinh⁡(⊖)=±isech(ζ),(38)

and

cosh⁡(⊖)=±coth(ζ),orcosh⁡(⊖)=±tanh(ζ),(39)

where i=−1.

We now compute the balance number of Eq. (3), which leads to n=1. Then the trail solution Eq. (35) in the EShGEEM takes the form,

Φ(⊖)=L1sinh⁡(⊖)+M1cosh⁡(⊖)+M0.(40)

Putting Eq. (40) into Eq. (3) along with Eq. (36), we obtain a polynomial of sinh⁡(ζ) and cosh⁡(ζ) functions, whose equating coefficients lead to a system of equations, and the solutions of the system of equations yield the following constraints:

Set 1:δ=−Γ12(k2b2+ka2−1)(2kb2+a2),w=k(εM12+k2b1+ϑM12+ka1)(k2b2+ka2−1),M0=0,L1=0,M1const.,(41)

Set 2:δ=−Γ2(k2b2+ka2−1)(2kb2+a2),w=k(εM12+k2b1+ϑM12+ka1)k2b2+ka2−1,M0=0,L1=M1,M1const.,(42)

Set3:δ=−Γ2(k2b2+ka2−1)(2kb2+a2),w=k(εM12+k2b1+ϑM12+ka1)k2b2+ka2−1,M0=0,L1=−M1,(43)

where M1 presents arbitrary constant and Γ1=(εk3M12b2+k3ϑM12b2+εk2M12a2+k2ϑM12a2+2εkM12b2−4k3b1b2+2kϑM12b2−εkM12−6k2a2b1−kϑM12−2ka1a2+6b1k+2a1), and Γ2=2εk3M12b2+2k3ϑM12b2+2εk2M12a2+2k2ϑM12a2+εkM12b2−2k3b1b2+kϑM12b2−2εkM12−3k2a2b1−2kϑM12−ka1a2+3kb1+a1.

Now for the Set 1, if we combine Eq. (41) with Eqs. (38)–(40), and substituting into Eq. (2), we obtain the exact soliton solutions of Eq. (1). These solutions of the model give us the optical shock wave and optical singular shock solitons,

Ψ8,9(x,t)=±M1tanh⁡(x−δt)ei(−kx+wt+ρ),(44)

Ψ10,11(x,t)=±M1coth⁡(x−δt)ei(−kx+wt+ρ),(45)

where δ and w come from Eq. (41). In the Eq. (44), Φ(x,t) comes in terms of tanh-function which presents kink shock waves. This solution presents a kink shock wave for the positive taking sign in the results (see Fig. 8a) but taking negative sign it presents an anti-kink shock wave (see Fig. 8b). The corresponding optical solitons represent dark-bell type peaked soliton (see Fig. 8c). On the other hand, in the Eq. (45), Φ(x,t) comes in terms of coth-function which presents kink shock waves with singularities i.e., singular kink type shock wave, for the positive sign it expresses singular kink but for the negative sign, it presents singular anti-kink type shock wave. The corresponding optical solitons represent bright peaked solitons with singularities (see Fig. 8d).

Figure 8: (a) Shock wave (Kink) for + via (tanh) of Eq. (44), (b) Shock wave (anti-kink) for−via (tanh) of Eq. (44), (c) Optical dark peakon via (tanh) of Eq. (44) and (d) Optical bright peaked (anti-peakon) with singularities Eq. (45) for the parametric values a1=5,a2=b1=b2=h=2,b1=δ=k=ε=1,w=0.5,ϑ=−2,L1=2,M1=1

Similarly, for the Set 2, if we combine Eq. (42) with Eqs. (38)–(40), and substituting into Eq. (2), we obtain the exact soliton solutions of Eq. (1). These solutions give us the combination of chirp-free bright and optical shock wave double solitons, and the combination of optical singular double solitons,

Ψ12,13(x,t)=M1[±isech(x−δt)±tanh⁡(x−δt)]ei(−kx+wt+ρ),(46)

Ψ14,15(x,t)=M1[±csch(x−δt)±coth⁡(x−δt)]ei(−kx+wt+ρ),(47)

where δ and w come from Eq. (42). And for the Set 3, if we combine Eq. (43) with Eqs. (38)–(40), and substituting into Eq. (2), we obtain the exact soliton solutions of Eq. (1). These solutions are able to give the combination of chirp-free bright and optical shock wave double solitons, and the combination of optical singular double solitons,

Ψ16,17(x,t)=M1[∓isech(x−δt)±tanh⁡(x−δt)]ei(−kx+wt+ρ),(48)

Ψ18,19(x,t)=M1[∓csch(x−δt)±coth⁡(x−δt)]ei(−kx+wt+ρ),(49)

where δ and w come from Eq. (43). The nature of solutions Ψ12,13(x,t) and Ψ16,17(x,t) are similar as they comes from produce periodic exponential function with linear combinations of sech-function (give bell solitonic nature) and tanh-function (give kink shock solitonic nature). Thus, the resulting nature of all the solutions is periodic bell wave with at least one kink in the surface depicted by the real part of Ψ17(x,t) in Fig. 9a and square of modulus of Ψ12(x,t) in Fig. 9b. We see that the square of modulus give us many small amplitude waves which can propagate to transmit signal rapidly through optical fiber. Besides this, the solutions Ψ14,15(x,t) have the same properties that they exhibits muti-peaked optical solitonic nature in presence of singularities. This character is displayed in Fig. 9c via |Ψ14(x,t)|2. The solutions Ψ18,19(x,t) have the same properties that they exhibits dark bell optical soliton in presence of singularities. This character is displayed in Fig. 9d via |Ψ18(x,t)|2.

Figure 9: (a) Combine bell kink soliton comes from real part of |Ψ17(x,t)|, (b) Small amplitude response |Ψ12(x,t)|2, (c) Multi-peaked optical soliton with singularities via Ψ14(x,t) and (d) Dark bell optical soliton with singularities via Ψ18(x,t) for the parametric values a1=5,a2=b2=2,b1δ=k=ε=M1=1,w=0.5,ϑ=−2

2.5 Optical Soliton Solutions to the Biswas-Arshed Model via the GK Method

The GK method [30] is a unique approach to obtaining the generalized solitons and periodic rogue waves of the nonlinear evolution equations (NLEEs) [20,21]. We now consider a rational series in this method as:

Ψ(ζ)=∑r=0nMr(F(ζ))r∑l=0mLl(F(ζ))l,(50)

where Mr,Ll, are constants to be later calculated and Mn≠0,Lm≠0. The function F(ζ) satisfies the Ricatti equation,

F′(ζ)=F2(ζ)−F(ζ),(51)

with the solution

F(ζ)=11+heζ,(52)

where h is the integral constant.

The trail solution Eq. (50) to the BAM takes the following form for the balance numbers n=2 and m=1,

Φ(ζ)=M0+M1F(ζ)+M2(F(ζ))2L0+L1F(ζ).(53)

Putting Eq. (53) into Eq. (3) along with Eq. (51), we obtain a polynomial of F(ζ) functions, whose equating coefficients lead to a system of equations, and provide the following set of constraints:

Set 1:δ=−Λ1L12(2k3b22+3k2a2b2+ka22−2kb2−a2),w=k(k2L12b1+kL12a1+εM12+ϑM12)L12(k2b2+ka2−1),M0=0,L0=0,M2=−2M1,(54)

where L1, M1 are constants and Λ1=2εk3M12b2+2k3ϑM12b2−2k3L12b1b2+2εk2M12a2+2k2ϑM12a2−3k2L12a2b1+εkM12b2+kϑM12b2−kL12a1a2−2εkM12−2kϑM12+3kL12b1+L12a1.

Set2:δ=−Λ28L02(k2b2+ka2−1)(2kb2+a2),w=k(8k2L02b1+8kL02a1−εM12−ϑM12)8L02(k2b2+ka2−1),M0=0,M2=−M1,L1=−2L0,(55)

where L0, M1 are constants and Λ2=εk3M12b2+k3ϑM12b2−16k3L02b1b2+εk2M12a2+k2ϑM12a2−24k2L02a2b1−εkM12b2−kϑM12b2−8kL02a1a2−εkM12−kϑM12+24kL02b1+8L02a1.

Set3:δ=−Λ32(2k3b22+3k2a2b2+ka22−2kb2−a2)L02,w=k(k2L02b1+kL02a1+εM02+ϑM02)L02(k2b2+ka2−1),M1=−2M0,M2=2M0,L1=−2L0,(56)

where L0, M0 are constants and Λ3=εk3M02b2+k3ϑM02b2−4k3L02b1b2+εk2M02a2+k2ϑM02a2−6k2L02a2b1+2εkM02b2+2kϑM02b2−2kL02a1a2−εkM02−kϑM02+6kL02b1+2L02a1.

Set4:δ=−Λ4(2k3b22+3k2a2b2+ka22−2kb2−a2)L02,w=k(k2L02b1+kL02a1+εM02+ϑM02)L02(k2b2+ka2−1),M1=M0(L1−2L0)L0,M2=−M0L1L0,(57)

where L0, L1, M0 are constants and Λ4=2εk3M02b2+2k3ϑM02b2−2k3L02b1b2+2εk2M02a2+2k2ϑM02a2−3k2L02a2b1+εkM02b2+kϑM02b2−kL02a1a2−2εkM02−2kϑM02+3kL02b1+L02a1.

Now for the Set 1, if we combine Eq. (54) with Eq. (53) and substituting into Eq. (2), we obtain the exact soliton solutions of Eq. (1). These solutions are able to give us optical rogue wave solitons,

Ψ20(x,t)=M1L1F(x−δt)ei(−kx+wt+ρ),(58)

where δ and w come from Eq. (54) and F(x−δt) comes from Eq. (52).

The behavior of the solution Ψ20(x,t) via Eq. (58) comes from produce periodic exponential function and exponential function (give solitonic nature). Thus, the resulting nature of the solution is periodic wave with a single shock (anti-kink) wave in both the real and imaginary parts, illustrated in Figs. 10b and 10c. But the simple Φ(x,t) and square of modulus of Ψ20(x,t) exhibits anti-kink type shock wave response in Figs. 10a and 10b, respectively.

Figure 10: (a) Shock wave, (b) Real part of, (c) Imaginary part of and (d) Shock wave of Eq. (58) for the parametric values a1=a2=b1=δ=k=ε=1,b2=h=2,w=0.5,ϑ=−2,L1=2,M1=1

Similar, for the Set 2, if we combine Eq. (55) with Eq. (53) and substituting into Eq. (2), we obtain the exact soliton solutions of Eq. (1). These solutions lead to optical rogue wave solitons,

Ψ21(x,t)=M1F(x−δt)(1−F(x−δt))L0{1−2F(x−δt)}ei(−kx+wt+ρ),(59)

where δ and w come from Eq. (55) and F(x−δt) from Eq. (52). For the Set 3, if we combine Eq. (56) with Eq. (53) and substituting into Eq. (2), we obtain the optical rogue wave soliton solutions of Eq. (1),

Ψ22(x,t)=M0{1−2F(x−δt)+2(F(x−δt))2}L0{1−2F(x−δt)}ei(−kx+wt+ρ),(60)

where δ and w come from Eq. (56), and F(x−δt) from Eq. (52). And for the Set 4, combining Eq. (57) with Eq. (53) and substituting into Eq. (2), we obtain the optical rogue wave soliton solutions of Eq. (1),

Ψ23(x,t)=M0L0+M0(L1−2L0)F(x−δt)−M0L1(F(x−δt))2L02{1−2F(x−δt)}ei(−kx+wt+ρ),(61)

where δ and w come from Eq. (57) and F(x−δt) from Eq. (52). The behavior of the solution Ψ21(x,t), Ψ22(x,t) and Ψ23(x,t) are same with singularities arises for L0=0 or {1−2F(x−δt)}=0. Since the resulting solution via Eqs. (59)–(61) comes from produce periodic exponential function and exponential function (give solitonic nature). Here, we reveal the results of Eq. (61) only against the three solutions, which exhibits a periodic wave with a singular shock wave in the both real and imaginary parts, illustrated in Fig. 11b and 11c. But the simple Φ(x,t) presents singular kink type shock wave (Fig. 11a) and the square of modulus of Ψ23(x,t) exhibits multi-peaked optical soliton with more singularities (Fig. 11d).