# Different Physical Structures of Solutions for a Generalized Resonant Dispersive Nonlinear Schrödinger Equation with Power Law Nonlinearity

Authors: K. S. Al-Ghafri
Journal: Journal of Applied Mathematics
 (2019)

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## Abstract

In this work, we investigate various types of solutions for the generalised resonant dispersive nonlinear Schrödinger equation (GRD-NLSE) with power law nonlinearity. Based on simple mathematical techniques, the complicated form of the GRD-NLSE is reduced to an ordinary differential equation (ODE) which has a variety of solutions. The analytic solution of the resulting ODE gives rise to bright soliton, singular soliton, peaked soliton, compacton solutions, solitary pattern solutions, rational solution, Weierstrass elliptic periodic type solutions, and some other types of solutions. Constraint conditions for the existence of solitons and other solutions are given.


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## Body

## 1. Introduction

Solitons have become one of the more attractive topics in the physical and natural science. The reason of this remarkable importance is that this type of nonlinear waves has many applications in the study of nonlinear optics, plasma physics, fluid dynamics, and several other disciplines [1–24]. For example, solitons transport information through optical fibers over transcontinental and transoceanic distances in a matter of a few femtoseconds. Furthermore, they also appear in Bose-Einstein condensates, α-helix proteins in clinical sciences, nuclear physics, and several others. The governing equation of such model is the nonlinear Schrödinger equation (NLSE).The formation of solitons in nonlinear optics, for example, is mainly due to a delicate balance between dispersion and nonlinearity in a model of NLSE. To analyse the dynamics of solitons, it is worthwhile to focus deeply on one model of the NLS family of equations with higher order nonlinear terms. There are many powerful mathematical tools that have been developed to study the behaviour of solitons in a medium dominated by NLSE. For more details, see [25–31]. In the present work, we will shed light on the study of the generalised resonant dispersive nonlinear Schrödinger equation (GRD-NLSE) with power law nonlinearity.The model of GRD-NLSE which is studied in the current paper has the form(1)iψn-1ψt+αψn-1ψxx+βψmψ+γψnxxψψ=0,where α,β, and γ are real-valued constants. The dependent variable ψ(x,t) is a complex-valued wave profile. Recently, the GRD-NLSE has been studied by many authors to examine the behaviour of solutions. Several integration schemes have been implemented to construct exact solutions such as ansatz method [13, 14], semi-inverse variational principle [15], simplest equation approach [16], first integral method [17], functional variable method, sine–cosine function method [18], (G′/G)-expansion method [19], trial solution approach [20], generalised extended tanh method [21], modified simple equation method [22], and improved extended tanh-equation method [23].In this paper, we aim to investigate the solitons and other types of solutions to GRD-NLSE. To achieve our goal, simple integration schemes will be applied to reduce the complicated form of GRD-NLSE to an ODE possessing various types of solutions. Solving the resulting ODE yields different physical structures of solutions for GRD-NLSE such as bright soliton, singular soliton, peaked soliton, compacton solutions, solitary pattern solutions, rational solution, Weierstrass elliptic periodic type solutions, and some other types of solutions.In the following section, (1) will be simplified to an ODE and then different types of exact solutions to this ODE will be extracted.


## 2. Mathematical Analysis and Solutions

In order to deal with the complicated form of the GRD-NLSE given by (1), we assume the travelling wave solution of the form(2)ψx,t=Uξei-κx+ωt+θ,ξ=x+2καt.Hence, (1) reduces to the following ordinary differential equation:(3)α+γUn′′-ω+ακ2Un+βUm+1=0,where prime denotes the derivative with respect to ξ. Setting V=Un, (3) becomes in the form(4)α+γV′′-ω+ακ2V+βVm+1/n=0. Now, multiplying by V′ and integrating once yield the following first-order ODE:(5)α+γdVdξ2=ω+ακ2V2-2nβm+n+1Vm+n+1/n,where the constant of integration is taken to be zero.


### 2.1. Solitary Wave Solution

Here, we aim to obtain the solitary wave solution of (1). Therefore, separating variables and integrating (5) give(6)∫dVV1-2nβ/m+n+1ω+ακ2Vm-n+1/n=ω+ακ2α+γξ, which leads to(7)2nn-m-1tanh-11-2nβm+n+1ω+ακ2Vm-n+1/n=ω+ακ2α+γξ.Equation (7) can be manipulated to yield(8)Vξ=Asech2Bξn/m-n+1, which represents a solitary wave with the amplitude(9)A=m+n+1ω+ακ22nβn/m-n+1,and the inverse width(10)B=n-m-12nω+ακ2α+γ,where(11)ω+ακ2α+γ>0.Eventually, the nontopological 1-soliton solution with power law nonlinearity to (1) is given by(12)ψx,t=A0sech2Bx+2καt1/m-n+1ei-κx+ωt+θ, where the amplitude, A0=A1/n, and width of the ψ profile are given by (9) and (10), respectively. The constraint given by (11) must stay valid in order for the soliton solution to exist.


### 2.2. Peakon Solution

In this subsection, we intend to find the peaked soliton of (1). Hence, we substitute the peakon assumption(13)Vξ=pe-rξ,into (5) and solve the resulting equation to find that(14)r=ω+ακ2-βα+γ,with the constraint m=n-1 and p can be any selective real number. Eventually, the peakon solution to (1) is given by(15)ψx,t=pe-ω+ακ2-β/α+γx+2καt1/nei-κx+ωt+θ. In case p is a negative constant so another type of peakon is presented, namely, antipeakon.Next, replacing the constantp by psign⁡(ξ) in (13) and substituting into (5) give a new solution to (1) called shock-peakon solution which can be written in the form(16)ψx,t=psign⁡ξe-ω+ακ2-β/α+γξ1/nei-κx+ωt+θ, where ξ=x+2καt and sign(ξ)=ξ/ξ. This type of peakons is a discontinuous wave so it is a shock wave. As we can see from the computation it has a discontinuous first-order derivative at ξ=0.


### 2.3. Compacton and Solitary Pattern Solutions

In order to obtain compacton and solitary patterns solutions of (1) we multiply (5) by V-(m+n+1)/n to arrive at(17)α+γV-m+n+1/2ndVdξ2=ω+ακ2Vn-m-1/n-2nβm+n+1,from which we find(18)α+γ2nn-m-12dVn-m-1/2ndξ2=ω+ακ2Vn-m-1/n-2nβm+n+1.Then, (18) can be simplified by assuming ϕ=V(n-m-1)/2n to obtain(19)dϕdξ2=1α+γn-m-12n2ω+ακ2ϕ2-2nβm+n+1.Solving (19), we obtain the following periodic type solutions:(20)ψx,t=A1cos2⁡B1x+2καt1/n-m-1ei-κx+ωt+θ,ψx,t=A1sin2⁡B1x+2καt1/n-m-1ei-κx+ωt+θ,which are valid when (ω+ακ2)(α+γ)<0, where(21)A1=2nβm+n+1ω+ακ2,B1=n-m-12n-ω+ακ2α+γ.In case of n>m+1, we arrive at the compacton solutions in the form(22)ψx,t=A1cos2⁡B1x+2καt1/n-m-1ei-κx+ωt+θ,B1x+2καt≤π2,ψx,t=0,otherwise, and(23)ψx,t=A1sin2⁡B1x+2καt1/n-m-1ei-κx+ωt+θ,B1x+2καt≤π,ψx,t=0,otherwise.Now, when the constant (ω+ακ2)(α+γ)>0, (19) admits the solitary pattern solutions(24)ψx,t=A1cosh2⁡B2x+2καt1/n-m-1ei-κx+ωt+θ,(25)ψx,t=-A1sinh2⁡B2x+2καt1/n-m-1ei-κx+ωt+θ,where A1 is give by (21) and(26)B2=n-m-12nω+ακ2α+γ.


### 2.4. Exponential Solution

In case ofm=n-1, then (5) will be reduced to(27)dVdξ2=ω+ακ2-βα+γV2.Separating variables and integrating (27) give(28)Vξ=ec±ω+ακ2-β/α+γξ,where c is the constant of integration. As a result, (1) possesses an exponential type solution in the form(29)ψx,t=ec±ω+ακ2-β/α+γx+2καt1/nei-κx+ωt+θ, when m=n-1.


### 2.5. Other Solutions

Now, we aim to extract more types of solutions to (1) using straightforward mathematical approach. Thus, we reduce (5) to a simple ODE by means of the following transformation. Letting(30)Vm-n+1/n=W2, we find(31)V=W2n/m-n+1,from which we reach(32)dV=2nm-n+1dW2n/m-n+1-1.Substituting (31) and (32) into (5) results in the following equation:(33)dWdξ2=1α+γm-n+12n2ω+ακ2W2-2nβm+n+1W4.Solving this equation, one can obtain the following types of solutions.


#### 2.5.1. Rational Type Solution

In case ofω=-ακ2, (1) admits the rational solution of the form(34)ψx,t=-2nm+n+1α+γβm-n+12x+2καt+c21/m-n+1ei-κx+ωt+θ,where c is the constant of integration.


#### 2.5.2. Complex Type Solution

The solution to (33) brings about the following complex solution for (1):(35)ψx,t=-m+n+1ω+ακ2nβ1ϵisinh⁡m-n+1/nω+ακ2/α+γx+2καt-11/m-n+1ei-κx+ωt+θ,where ϵ=±1 and i=-1. This solution demands(36)ω+ακ2α+γ>0.


#### 2.5.3. Bright Soliton Solutions

The solution to (33) leads to the following forms of bright solitons for (1):(37)ψx,t=m+n+1ω+ακ22nβsech2m-n+12nω+ακ2α+γx+2καt1/m-n+1ei-κx+ωt+θ.This soliton is valid for(38)ω+ακ2α+γ>0.ψx,t=asech2⁡x+2καtbsech2⁡x+2καt+ctanh⁡x+2καt-2b2/m-n+1ei-κx+ωt+θ, which is valid when(39)m-n+12n2ω+ακ2α+γ=4,m-n+12n22nβm+n+1α+γ=4b2-c2a2, where a,b,c are arbitrary constants.(40)ψx,t=m+n+1ω+ακ2nβ1cosh⁡m-n+1/nω+ακ2/α+γx+2καt+11/m-n+1ei-κx+ωt+θ.Solution (40) is valid when(41)ω+ακ2α+γ>0.


#### 2.5.4. Singular Soliton Solutions

The solution to (33) gives rise to the following forms of singular solitons for (1):(42)ψx,t=m+n+1ω+ακ22nβcsch2m-n+12nω+ακ2α+γx+2καt1/m-n+1ei-κx+ωt+θ.This soliton is valid for(43)ω+ακ2α+γ>0.ψx,t=acsch2⁡x+2καtbcsch2⁡x+2καt+ccoth⁡x+2καt+2b2/m-n+1ei-κx+ωt+θ, which is valid when(44)m-n+12n2ω+ακ2α+γ=4,m-n+12n22nβm+n+1α+γ=4b2-c2a2, where a,b,c are arbitrary constants. (45)ψx,t=-m+n+1ω+ακ2nβ1coshm-n+1/nω+ακ2/α+γx+2καt-11/m-n+1ei-κx+ωt+θ,Solution (45) is valid when(46)ω+ακ2α+γ>0.


#### 2.5.5. Singular Periodic Solutions

The solution to (33) provides the following variety of singular periodic solutions for (1).(47)ψx,t=m+n+1ω+ακ22nβsec2m-n+12n-ω+ακ2α+γx+2καt1/m-n+1ei-κx+ωt+θ,(48)ψx,t=m+n+1ω+ακ22nβcsc2m-n+12n-ω+ακ2α+γx+2καt1/m-n+1ei-κx+ωt+θ.Equations (47) and (48) demand(49)ω+ακ2α+γ<0.(50)ψx,t=asec2⁡x+2καtbsec2⁡x+2καt+ctan⁡x+2καt-2b2/m-n+1ei-κx+ωt+θ,(51)ψx,t=acsc2⁡x+2καtbcsc2⁡x+2καt+ccot⁡x+2καt-2b2/m-n+1ei-κx+ωt+θ.Both solutions (50) and (51) are valid for(52)m-n+12n2ω+ακ2α+γ=-4,m-n+12n22nβm+n+1α+γ=-4b2+c2a2, where a,b,c are arbitrary constants.(53)ψx,t=-m+n+1ω+ακ2nβ1ϵcos⁡m-n+1/n-ω+ακ2/α+γx+2καt-11/m-n+1ei-κx+ωt+θ,(54)ψx,t=-m+n+1ω+ακ2nβ1ϵsin⁡m-n+1/n-ω+ακ2/α+γx+2καt-11/m-n+1ei-κx+ωt+θ,where ϵ=±1. Solutions (53) and (54) imply that(55)ω+ακ2α+γ<0.


#### 2.5.6. Weierstrass Elliptic Periodic Type Solutions

The solution to (33) generates Weierstrass elliptic type solutions for (1) in the following forms:(56)ψx,t=-2nm+n+1α+γ3m-n+12β3℘x+2καt,g2,g3-m-n+12n2ω+ακ2α+γ1/m-n+1×ei-κx+ωt+θ,where the invariants of the Weierstrass elliptic function are given by(57)g2=43m-n+12n2ω+ακ2α+γ2,g3=-827m-n+12n2ω+ακ2α+γ3.(58)ψx,t=-2nm+n+1α+γm-n+12β3℘′x+2καt,g2,g36℘x+2καt,g2,g3+m-n+1/2n2ω+ακ2/α+γ21/m-n+1×ei-κx+ωt+θ,(59)ψx,t=m+n+1ω+ακ22nβ12℘x+2καt,g2,g3-m-n+1/2n2ω+ακ2/α+γ12℘x+2καt,g2,g3+5m-n+1/2n2ω+ακ2/α+γ21/m-n+1×ei-κx+ωt+θ,where both solutions (58) and (59) are valid for the invariants of the Weierstrass elliptic function given by(60)g2=112m-n+12n2ω+ακ2α+γ2,g3=-1216m-n+12n2ω+ακ2α+γ3.Overall, the majority of results obtained here for (1) are very new. In comparison with some previous studies, the solutions given by (34), (35), (37), (42), (47), (48), and (54) are already derived in [18, 23] while the rest of the solutions extracted here are new exact solutions.It should be noted that the proposed transformation in (31) has led to the ODE (33) which is simpler than that derived in [23]. Further to this, (33) can be converted into the form(61)d2Wdξ2=1α+γm-n+12n2ω+ακ2W-4nβm+n+1W3.This second-order equation is known to admit the application of many solution methods like the Jacobi elliptic function method [32], the exp-function method [33], the G′/G-expansion method [34], the generalised Kudryashov method [35], etc. As a result, a lot of exact analytic solutions can be constructed to the GRD-NLSE (1).


## 2.1. Solitary Wave Solution

Here, we aim to obtain the solitary wave solution of (1). Therefore, separating variables and integrating (5) give(6)∫dVV1-2nβ/m+n+1ω+ακ2Vm-n+1/n=ω+ακ2α+γξ, which leads to(7)2nn-m-1tanh-11-2nβm+n+1ω+ακ2Vm-n+1/n=ω+ακ2α+γξ.Equation (7) can be manipulated to yield(8)Vξ=Asech2Bξn/m-n+1, which represents a solitary wave with the amplitude(9)A=m+n+1ω+ακ22nβn/m-n+1,and the inverse width(10)B=n-m-12nω+ακ2α+γ,where(11)ω+ακ2α+γ>0.Eventually, the nontopological 1-soliton solution with power law nonlinearity to (1) is given by(12)ψx,t=A0sech2Bx+2καt1/m-n+1ei-κx+ωt+θ, where the amplitude, A0=A1/n, and width of the ψ profile are given by (9) and (10), respectively. The constraint given by (11) must stay valid in order for the soliton solution to exist.


## 2.2. Peakon Solution

In this subsection, we intend to find the peaked soliton of (1). Hence, we substitute the peakon assumption(13)Vξ=pe-rξ,into (5) and solve the resulting equation to find that(14)r=ω+ακ2-βα+γ,with the constraint m=n-1 and p can be any selective real number. Eventually, the peakon solution to (1) is given by(15)ψx,t=pe-ω+ακ2-β/α+γx+2καt1/nei-κx+ωt+θ. In case p is a negative constant so another type of peakon is presented, namely, antipeakon.Next, replacing the constantp by psign⁡(ξ) in (13) and substituting into (5) give a new solution to (1) called shock-peakon solution which can be written in the form(16)ψx,t=psign⁡ξe-ω+ακ2-β/α+γξ1/nei-κx+ωt+θ, where ξ=x+2καt and sign(ξ)=ξ/ξ. This type of peakons is a discontinuous wave so it is a shock wave. As we can see from the computation it has a discontinuous first-order derivative at ξ=0.


## 2.3. Compacton and Solitary Pattern Solutions

In order to obtain compacton and solitary patterns solutions of (1) we multiply (5) by V-(m+n+1)/n to arrive at(17)α+γV-m+n+1/2ndVdξ2=ω+ακ2Vn-m-1/n-2nβm+n+1,from which we find(18)α+γ2nn-m-12dVn-m-1/2ndξ2=ω+ακ2Vn-m-1/n-2nβm+n+1.Then, (18) can be simplified by assuming ϕ=V(n-m-1)/2n to obtain(19)dϕdξ2=1α+γn-m-12n2ω+ακ2ϕ2-2nβm+n+1.Solving (19), we obtain the following periodic type solutions:(20)ψx,t=A1cos2⁡B1x+2καt1/n-m-1ei-κx+ωt+θ,ψx,t=A1sin2⁡B1x+2καt1/n-m-1ei-κx+ωt+θ,which are valid when (ω+ακ2)(α+γ)<0, where(21)A1=2nβm+n+1ω+ακ2,B1=n-m-12n-ω+ακ2α+γ.In case of n>m+1, we arrive at the compacton solutions in the form(22)ψx,t=A1cos2⁡B1x+2καt1/n-m-1ei-κx+ωt+θ,B1x+2καt≤π2,ψx,t=0,otherwise, and(23)ψx,t=A1sin2⁡B1x+2καt1/n-m-1ei-κx+ωt+θ,B1x+2καt≤π,ψx,t=0,otherwise.Now, when the constant (ω+ακ2)(α+γ)>0, (19) admits the solitary pattern solutions(24)ψx,t=A1cosh2⁡B2x+2καt1/n-m-1ei-κx+ωt+θ,(25)ψx,t=-A1sinh2⁡B2x+2καt1/n-m-1ei-κx+ωt+θ,where A1 is give by (21) and(26)B2=n-m-12nω+ακ2α+γ.


## 2.4. Exponential Solution

In case ofm=n-1, then (5) will be reduced to(27)dVdξ2=ω+ακ2-βα+γV2.Separating variables and integrating (27) give(28)Vξ=ec±ω+ακ2-β/α+γξ,where c is the constant of integration. As a result, (1) possesses an exponential type solution in the form(29)ψx,t=ec±ω+ακ2-β/α+γx+2καt1/nei-κx+ωt+θ, when m=n-1.


## 2.5. Other Solutions

Now, we aim to extract more types of solutions to (1) using straightforward mathematical approach. Thus, we reduce (5) to a simple ODE by means of the following transformation. Letting(30)Vm-n+1/n=W2, we find(31)V=W2n/m-n+1,from which we reach(32)dV=2nm-n+1dW2n/m-n+1-1.Substituting (31) and (32) into (5) results in the following equation:(33)dWdξ2=1α+γm-n+12n2ω+ακ2W2-2nβm+n+1W4.Solving this equation, one can obtain the following types of solutions.


### 2.5.1. Rational Type Solution

In case ofω=-ακ2, (1) admits the rational solution of the form(34)ψx,t=-2nm+n+1α+γβm-n+12x+2καt+c21/m-n+1ei-κx+ωt+θ,where c is the constant of integration.


### 2.5.2. Complex Type Solution

The solution to (33) brings about the following complex solution for (1):(35)ψx,t=-m+n+1ω+ακ2nβ1ϵisinh⁡m-n+1/nω+ακ2/α+γx+2καt-11/m-n+1ei-κx+ωt+θ,where ϵ=±1 and i=-1. This solution demands(36)ω+ακ2α+γ>0.


### 2.5.3. Bright Soliton Solutions

The solution to (33) leads to the following forms of bright solitons for (1):(37)ψx,t=m+n+1ω+ακ22nβsech2m-n+12nω+ακ2α+γx+2καt1/m-n+1ei-κx+ωt+θ.This soliton is valid for(38)ω+ακ2α+γ>0.ψx,t=asech2⁡x+2καtbsech2⁡x+2καt+ctanh⁡x+2καt-2b2/m-n+1ei-κx+ωt+θ, which is valid when(39)m-n+12n2ω+ακ2α+γ=4,m-n+12n22nβm+n+1α+γ=4b2-c2a2, where a,b,c are arbitrary constants.(40)ψx,t=m+n+1ω+ακ2nβ1cosh⁡m-n+1/nω+ακ2/α+γx+2καt+11/m-n+1ei-κx+ωt+θ.Solution (40) is valid when(41)ω+ακ2α+γ>0.


### 2.5.4. Singular Soliton Solutions

The solution to (33) gives rise to the following forms of singular solitons for (1):(42)ψx,t=m+n+1ω+ακ22nβcsch2m-n+12nω+ακ2α+γx+2καt1/m-n+1ei-κx+ωt+θ.This soliton is valid for(43)ω+ακ2α+γ>0.ψx,t=acsch2⁡x+2καtbcsch2⁡x+2καt+ccoth⁡x+2καt+2b2/m-n+1ei-κx+ωt+θ, which is valid when(44)m-n+12n2ω+ακ2α+γ=4,m-n+12n22nβm+n+1α+γ=4b2-c2a2, where a,b,c are arbitrary constants. (45)ψx,t=-m+n+1ω+ακ2nβ1coshm-n+1/nω+ακ2/α+γx+2καt-11/m-n+1ei-κx+ωt+θ,Solution (45) is valid when(46)ω+ακ2α+γ>0.


### 2.5.5. Singular Periodic Solutions

The solution to (33) provides the following variety of singular periodic solutions for (1).(47)ψx,t=m+n+1ω+ακ22nβsec2m-n+12n-ω+ακ2α+γx+2καt1/m-n+1ei-κx+ωt+θ,(48)ψx,t=m+n+1ω+ακ22nβcsc2m-n+12n-ω+ακ2α+γx+2καt1/m-n+1ei-κx+ωt+θ.Equations (47) and (48) demand(49)ω+ακ2α+γ<0.(50)ψx,t=asec2⁡x+2καtbsec2⁡x+2καt+ctan⁡x+2καt-2b2/m-n+1ei-κx+ωt+θ,(51)ψx,t=acsc2⁡x+2καtbcsc2⁡x+2καt+ccot⁡x+2καt-2b2/m-n+1ei-κx+ωt+θ.Both solutions (50) and (51) are valid for(52)m-n+12n2ω+ακ2α+γ=-4,m-n+12n22nβm+n+1α+γ=-4b2+c2a2, where a,b,c are arbitrary constants.(53)ψx,t=-m+n+1ω+ακ2nβ1ϵcos⁡m-n+1/n-ω+ακ2/α+γx+2καt-11/m-n+1ei-κx+ωt+θ,(54)ψx,t=-m+n+1ω+ακ2nβ1ϵsin⁡m-n+1/n-ω+ακ2/α+γx+2καt-11/m-n+1ei-κx+ωt+θ,where ϵ=±1. Solutions (53) and (54) imply that(55)ω+ακ2α+γ<0.


### 2.5.6. Weierstrass Elliptic Periodic Type Solutions

The solution to (33) generates Weierstrass elliptic type solutions for (1) in the following forms:(56)ψx,t=-2nm+n+1α+γ3m-n+12β3℘x+2καt,g2,g3-m-n+12n2ω+ακ2α+γ1/m-n+1×ei-κx+ωt+θ,where the invariants of the Weierstrass elliptic function are given by(57)g2=43m-n+12n2ω+ακ2α+γ2,g3=-827m-n+12n2ω+ακ2α+γ3.(58)ψx,t=-2nm+n+1α+γm-n+12β3℘′x+2καt,g2,g36℘x+2καt,g2,g3+m-n+1/2n2ω+ακ2/α+γ21/m-n+1×ei-κx+ωt+θ,(59)ψx,t=m+n+1ω+ακ22nβ12℘x+2καt,g2,g3-m-n+1/2n2ω+ακ2/α+γ12℘x+2καt,g2,g3+5m-n+1/2n2ω+ακ2/α+γ21/m-n+1×ei-κx+ωt+θ,where both solutions (58) and (59) are valid for the invariants of the Weierstrass elliptic function given by(60)g2=112m-n+12n2ω+ακ2α+γ2,g3=-1216m-n+12n2ω+ακ2α+γ3.Overall, the majority of results obtained here for (1) are very new. In comparison with some previous studies, the solutions given by (34), (35), (37), (42), (47), (48), and (54) are already derived in [18, 23] while the rest of the solutions extracted here are new exact solutions.It should be noted that the proposed transformation in (31) has led to the ODE (33) which is simpler than that derived in [23]. Further to this, (33) can be converted into the form(61)d2Wdξ2=1α+γm-n+12n2ω+ακ2W-4nβm+n+1W3.This second-order equation is known to admit the application of many solution methods like the Jacobi elliptic function method [32], the exp-function method [33], the G′/G-expansion method [34], the generalised Kudryashov method [35], etc. As a result, a lot of exact analytic solutions can be constructed to the GRD-NLSE (1).


## 2.5.1. Rational Type Solution

In case ofω=-ακ2, (1) admits the rational solution of the form(34)ψx,t=-2nm+n+1α+γβm-n+12x+2καt+c21/m-n+1ei-κx+ωt+θ,where c is the constant of integration.


## 2.5.2. Complex Type Solution

The solution to (33) brings about the following complex solution for (1):(35)ψx,t=-m+n+1ω+ακ2nβ1ϵisinh⁡m-n+1/nω+ακ2/α+γx+2καt-11/m-n+1ei-κx+ωt+θ,where ϵ=±1 and i=-1. This solution demands(36)ω+ακ2α+γ>0.


## 2.5.3. Bright Soliton Solutions

The solution to (33) leads to the following forms of bright solitons for (1):(37)ψx,t=m+n+1ω+ακ22nβsech2m-n+12nω+ακ2α+γx+2καt1/m-n+1ei-κx+ωt+θ.This soliton is valid for(38)ω+ακ2α+γ>0.ψx,t=asech2⁡x+2καtbsech2⁡x+2καt+ctanh⁡x+2καt-2b2/m-n+1ei-κx+ωt+θ, which is valid when(39)m-n+12n2ω+ακ2α+γ=4,m-n+12n22nβm+n+1α+γ=4b2-c2a2, where a,b,c are arbitrary constants.(40)ψx,t=m+n+1ω+ακ2nβ1cosh⁡m-n+1/nω+ακ2/α+γx+2καt+11/m-n+1ei-κx+ωt+θ.Solution (40) is valid when(41)ω+ακ2α+γ>0.


## 2.5.4. Singular Soliton Solutions

The solution to (33) gives rise to the following forms of singular solitons for (1):(42)ψx,t=m+n+1ω+ακ22nβcsch2m-n+12nω+ακ2α+γx+2καt1/m-n+1ei-κx+ωt+θ.This soliton is valid for(43)ω+ακ2α+γ>0.ψx,t=acsch2⁡x+2καtbcsch2⁡x+2καt+ccoth⁡x+2καt+2b2/m-n+1ei-κx+ωt+θ, which is valid when(44)m-n+12n2ω+ακ2α+γ=4,m-n+12n22nβm+n+1α+γ=4b2-c2a2, where a,b,c are arbitrary constants. (45)ψx,t=-m+n+1ω+ακ2nβ1coshm-n+1/nω+ακ2/α+γx+2καt-11/m-n+1ei-κx+ωt+θ,Solution (45) is valid when(46)ω+ακ2α+γ>0.


## 2.5.5. Singular Periodic Solutions

The solution to (33) provides the following variety of singular periodic solutions for (1).(47)ψx,t=m+n+1ω+ακ22nβsec2m-n+12n-ω+ακ2α+γx+2καt1/m-n+1ei-κx+ωt+θ,(48)ψx,t=m+n+1ω+ακ22nβcsc2m-n+12n-ω+ακ2α+γx+2καt1/m-n+1ei-κx+ωt+θ.Equations (47) and (48) demand(49)ω+ακ2α+γ<0.(50)ψx,t=asec2⁡x+2καtbsec2⁡x+2καt+ctan⁡x+2καt-2b2/m-n+1ei-κx+ωt+θ,(51)ψx,t=acsc2⁡x+2καtbcsc2⁡x+2καt+ccot⁡x+2καt-2b2/m-n+1ei-κx+ωt+θ.Both solutions (50) and (51) are valid for(52)m-n+12n2ω+ακ2α+γ=-4,m-n+12n22nβm+n+1α+γ=-4b2+c2a2, where a,b,c are arbitrary constants.(53)ψx,t=-m+n+1ω+ακ2nβ1ϵcos⁡m-n+1/n-ω+ακ2/α+γx+2καt-11/m-n+1ei-κx+ωt+θ,(54)ψx,t=-m+n+1ω+ακ2nβ1ϵsin⁡m-n+1/n-ω+ακ2/α+γx+2καt-11/m-n+1ei-κx+ωt+θ,where ϵ=±1. Solutions (53) and (54) imply that(55)ω+ακ2α+γ<0.


## 2.5.6. Weierstrass Elliptic Periodic Type Solutions

The solution to (33) generates Weierstrass elliptic type solutions for (1) in the following forms:(56)ψx,t=-2nm+n+1α+γ3m-n+12β3℘x+2καt,g2,g3-m-n+12n2ω+ακ2α+γ1/m-n+1×ei-κx+ωt+θ,where the invariants of the Weierstrass elliptic function are given by(57)g2=43m-n+12n2ω+ακ2α+γ2,g3=-827m-n+12n2ω+ακ2α+γ3.(58)ψx,t=-2nm+n+1α+γm-n+12β3℘′x+2καt,g2,g36℘x+2καt,g2,g3+m-n+1/2n2ω+ακ2/α+γ21/m-n+1×ei-κx+ωt+θ,(59)ψx,t=m+n+1ω+ακ22nβ12℘x+2καt,g2,g3-m-n+1/2n2ω+ακ2/α+γ12℘x+2καt,g2,g3+5m-n+1/2n2ω+ακ2/α+γ21/m-n+1×ei-κx+ωt+θ,where both solutions (58) and (59) are valid for the invariants of the Weierstrass elliptic function given by(60)g2=112m-n+12n2ω+ακ2α+γ2,g3=-1216m-n+12n2ω+ακ2α+γ3.Overall, the majority of results obtained here for (1) are very new. In comparison with some previous studies, the solutions given by (34), (35), (37), (42), (47), (48), and (54) are already derived in [18, 23] while the rest of the solutions extracted here are new exact solutions.It should be noted that the proposed transformation in (31) has led to the ODE (33) which is simpler than that derived in [23]. Further to this, (33) can be converted into the form(61)d2Wdξ2=1α+γm-n+12n2ω+ακ2W-4nβm+n+1W3.This second-order equation is known to admit the application of many solution methods like the Jacobi elliptic function method [32], the exp-function method [33], the G′/G-expansion method [34], the generalised Kudryashov method [35], etc. As a result, a lot of exact analytic solutions can be constructed to the GRD-NLSE (1).


## 3. Discussion and Conclusion

This study scoped different physical structures of solutions for GRD-NLSE with power law nonlinearity. Applying a simple mathematical scheme allowed us to simplify the complex form of GRD-NLSE to an ODE. It is found that the constructed ODE is rich in various types of solitons and other solutions for GRD-NLSE. The derived solutions include bright soliton, singular soliton, peaked soliton, compacton solutions, solitary pattern solutions, rational solution, trigonometric function solutions, and Weierstrass elliptic periodic type solutions. All generated solutions are verified by utilising symbolic computation. The results obtained here can be useful to understand the physics of nonlinear optical fibers.



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*Source: 6143102-2019-02-03.xml*