# Classification of All Single Traveling Wave Solutions of Fractional Perturbed Gerdjikov–Ivanov Equation

Authors: Zhao Li; Tianyong Han
Journal: Mathematical Problems in Engineering
 (2021)

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## Abstract

The fractional perturbed Gerdjikov–Ivanov (pGI) equation plays a momentous role in nonlinear fiber optics, especially in the application of photonic crystal fibers. Constructing traveling wave solutions to this equation is a very challenging task in physics and mathematics. In the current article, our main purpose is to give the classifications of traveling wave solutions of the fractional pGI equation. These results can help physicists to further explain the complex fractional pGI equation.


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## Body

## 1. Introduction

The nonlinear Schrödinger equation (NLSE) is usually used to describe many problems in nonlinear optics, quantum mechanics, fluid dynamics, plasmas, and so on [1–5]. Among them, NLSE with cubic nonlinearity, namely, the perturbed Gerdjikov-Ivanov (pGI) equation, is the most studied [6–11], which is given by(1)i∂u∂t+a∂2u∂x2+bu4u=icu2∂u¯∂x+δ1∂u∂x+δ2∂u2u∂x+θ∂u2∂xu,where ut,x is a complex valued wave profile with respect to time and space variables. u¯ is the complex conjugation of u. c stands for the nonlinear dispersion coefficient. δ1, δ2, and θ are known parameters.In recent years, with the development of fractional derivative, the fractional perturbed Gerdjikov–Ivanov equation has been widely concerned by many researchers in the following form:(2)iDtαu+aDx2αu+bu4u=icu2Dxαu¯+δ1Dxαu+δ2Dxαu2u+θDxαu2u,0<α<1,where u=ut,x represents unknown functions about time t and space x, respectively. u¯ is the complex conjugation of u. Dtαu and Dxαu are the conformable fractional derivative [12–14].In recent years, the traveling wave solution of the fractional pGI equation has attained a lot of interest in the filed of nonlinear science. Some explicit traveling wave solutions of the fractional pGI equation have been reported [15–17]. In Reference [15], Li and his collaborators studied the space-time fractional pGI equation by the fractional H-expansion method. In Reference [16], Ghanbari and Dumitru obtained the optical solutions of the fractional GI equation by using the generalized exponential rational function method. In Reference [17], Zulfiqar and Ahmad explored the optical solution of the fractional pGI equation by the Tanh method and Tanh-coth method, respectively. The main purpose of this paper is to give the classifications of traveling wave solutions of the fractional pGI equation by using the complete discrimination system method for polynomial, which is an important method to find the exact traveling wave solution of the fractional partial differential equation (FPDE). In general, the FPDE can be simplified to the following complete discriminant system by traveling wave transformation:(3)Ψ′ξ=FΨ,ϕ1,…,ϕm,where ϕ1,…,ϕm are parameters. FΨ,ϕ1,…,ϕm is third-order polynomial, fourth-order polynomial, or fifth-order polynomial. Then, integrating equation (3) once, we obtain(4)ξ−ξ0=∫dΨFΨ,ϕ1,…,ϕm,where ξ0 is the integration constant. The readers can refer to [2, 4, 18] for details of this method.The structure of this paper is as follows. In Section2, we simplify equation (2) to the nonlinear ordinary differential equation by fractional traveling wave transformation. Then, the complete discrimination system is used to construct the classification of all single traveling wave solutions of the fractional pGI equation. In Section 3, we give a summary.


## 2. Traveling Wave Solutions of System (2)

Consider the fractional traveling wave transformation as follows:(5)ut,x=Uξeiη,ξ=xαα−vtαα,η=−kxαα+wtαα,where v is the phase component, w represents the wavenumber, and k is the frequency.Substituting (5) into (2) and separating the real and imaginary part, we obtain(6)aU″−w+ak2+δ1kU+kc−δ2U3+bU5=0,and(7)v=−δ1−2ak,θ=−12c+3δ2.Multiplying equation (6) by U′ and integrating it with respect to ξ, we obtain(8)a2U′2+b6U6+kc−δ24U4−w+ak2+kδ12U2+d=0,where d is the integral constant. Next, we fix U=ψ, then equation (8) can be converted into(9)ψ′2=−4b3aψ4−2kc−δ2aψ3+4w+ak2+kδ1aψ2−8daψ.(1)
Ifd=0, equation (9) can be simplified as(10)ψ′2=ψ2b3ψ2+b2ψ+b1,whereb3=−4b/3a, b2=−2kc−δ2/a, and b1=4w+ak2+kδ1/a. Then, equation (10) can be written in the following integral form:(11)∫dψψb3ψ3+b2ψ+b1=±2ξ−ξ0.DenoteΔ=b22−4b1b3, where Δ is the discriminant of polynomial b3ψ3+b2ψ+b1. According to the complete discriminant method of second-order polynomials, the classification of all solutions of equation (11) has the following three cases:Case 1.1. Suppose that Δ=0. From equation (11), we can get(12)±b2b3b3ξ−ξ0=lnψ−b2/2b3ψ.Case 1.2. Suppose that Δ>0. When b3>0, we have from (11)(13)±b3ξ−ξ0=1γ1γ2ln−γ2ψ−γ1−−γ1ψ−γ22ψ,±b3ξ−ξ0=1γ1γ2lnγ2ψ−γ1−γ1ψ−γ22ψ,±b3ξ−ξ0=1−γ1γ2arcsin−γ2ψ−γ1+−γ1ψ−γ2ψγ1−γ2.Whenb3<0, we have from (11)(14)±−b3ξ−ξ0=1−γ1γ2ln−γ2ψ+γ1−γ1ψ−γ22ψ,±−b3ξ−ξ0=1−γ1γ2lnγ2−ψ+γ1−−γ1ψ−γ22ψ,±−b3ξ−ξ0=1γ1γ2arcsin−γ2ψ+γ1−γ1ψ−γ22ψ,whereγ1=−b2+Δ/2b3 and γ2=−b2−Δ/2b3.Case 1.3. Suppose that Δ<0. Then, we have(15)±b1ξ−ξ0=1b12ln−1/2b2b1ψ+b1−b3ψ2−b2ψ+b1ψ,b1>0.(2)
Ifd≠0, equation (9) can be simplified as(16)ψ′2=ψ−4b3aψ3−2kc−δ2aψ2+4w+ak2+kδ1aψ−8da.Then, equation (16) can be written in the following integral form:(17)∫dψεψψ3+pψ2+qψ+r=±2−bε3aξ−ξ0,whereε=±1, p=3b/kc−δ, q=−3w+ak2+kδ1/b, and r=6d/b.Denote(18)Δ=−272p327+r−qr32−4q−p233,D1=q−p23,whereΔ is the discriminant of polynomial Gψ=ψ3+pψ2+qψ+r. According to the complete discriminant method of third-order polynomials, the classification of all solutions of equation (11) has the following four cases:Case 2.1. Assume that Δ=0, D1<0. Denote Gψ=ψ−β12ψ−β2, where β1≠β2 and β2>0.Ifε=1, ab<0, β1>β2, and ψ>β2 or β1<0 and ψ<0, then the implicit solution of equation (17) is(19)±2−b3aξ−ξ0=1β1β1−β2lnβ1ψ−β2−ψβ1−β22ψ−β1.Ifε=1, ab<0, β1>β2, and ψ<0 or β1<0 and ψ<β2, then the implicit solution of equation (17) is(20)±2−b3aξ−ξ0=1β1β1−β2ln−β1ψ−β2−ψβ2−β12ψ−β1.Ifε=1, ab<0, and β2>β1>0, then the implicit solution of equation (17) is(21)±2−b3aξ−ξ0=1β1β2−β1arcsinβ1ψ−β2+ψβ1−β2β2ψ−β1.Ifε=−1, ab>0, β1>β2, and ψ>β2 or β1<0 and ψ<0, then the implicit solution of equation (17) is(22)±2b3aξ−ξ0=1β1β2−β1lnβ1−ψ+β2−ψβ1−β22ψ−β1.Ifε=−1, ab>0, β1>β2, and ψ<0 or β1<0 and ψ<β2, then the implicit solution of equation (17) is(23)±2b3aξ−ξ0=1β1β2−β1ln−β1−ψ+β2−ψβ1−β22ψ−β1.Ifε=−1, ab>0, and β2>β1>0, then the implicit solution of equation (17) is(24)±2b3aξ−ξ0=1β1β1−β2arcsinβ1−ψ+β2+ψβ2−β1β2ψ−β1.Case 2.2. Assume that Δ=0 and D1=0. Denote Gψ=ψ−β13, where β1 is the real constant.Ifε=1, ψ>β1, and ψ>0 or ψ<β1 and ψ<0 then the explicit solution of equation (2) is(25)u1t,x=±β1−b/3aβ12xα/α−vtα/α−1+β11/2ei−kxα/α+wtα/α.Ifε=−1, ψ>β1, and ψ<0 or ψ<β1 and ψ>0, then the explicit solution of equation (2) is(26)u2t,x=±β1b/3aβ12xα/α−vtα/α−1+β11/2ei−kxα/α+wtα/α.Case 2.3. Assume that Δ>0 and D1<0. Denote Gψ=ψ−β1ψ−β2ψ−β3, where 0<β1<β2<β3.Ifε=1, ψ<0, or ψ>β3, then we make transformation ψ=β2β3sin2ϑ−β2β3/β3sin2ϑ−β2, and we obtain(27)±−b3aξ−ξ0=1β2β3−β1∫dϑ1−m12sin2ϑ,wherem12=β3β1−β2/β2β1−β3. Then, the explicit solution of equation (2) is(28)u3t,x=±β2β3sn2−b/3aβ2β3−β1xα/α−vtα/α,m1−β2β3β3sn2−b/3aβ2β3−β1xα/α−vtα/α,m1−β21/2ei−kxα/α+wtα/α.Ifε=1 and β1<ψ<β2, then we make transformation ψ=−β1β2/β2−β1sin2ϑ−β2. Similarly, the explicit solution of equation (2) is(29)u4t,x=±−β1β2β2−β1sn2−b/3aβ2β3−β1xα/α−vtα/α,m1−β21/2ei−kxα/α+wtα/α.Ifε=−1 and β2<ψ<β3, then we make transformation ψ=β1β3−β2sin2ϑ−β2β3−β1/β3−β2sin2ϑ−β3−β1, and the explicit solution of equation (2) is(30)u5t,x=±β1β3−β2sn2−b/3aβ2β3−β1xα/α−vtα/α,m2−β2β3−β1β3−β2sn2−b/3aβ2β3−β1xα/α−vtα/α,m2−β3−β11/2ei−kxα/α+wtα/α,wherem22=β1β2−β3/β2β1−β3.Ifε=−1 and 0<ψ<β1, then we make transformation ψ=β1β3sin2ϑ/β1sin2ϑ−β1−β3. Similarly, the explicit solution of equation (2) is(31)u6t,x=±β1β2sn2xα/α−vtα/α/2b/3aβ2β3−β1,m2β1sn2xα/α−vtα/α/2b/3aβ2β3−β1,m2−β1−β31/2ei−kxα/α+wtα/α.Case 2.4. Assume that Δ<0. Denote Gψ=ψ−β1ψ−β22+γ, where β1,β2,γ>0 are real constants. Then, we make transformation ψ=d1cosϑ+d2/d3cosϑ+d4, where d1=β1/2d3−d4, d2=−β1/2d3−d4, d3=β1−β2−γ/m, d4=β1−β2−γm, m=D±D2+1, and D=γ2−β2β1−β2/β2γ.Ifε=1 and m=D−D2+1, then the explicit solution of equation (2) is(32)u7t,x=±d1cnxα−vtα/α2b/3aγmβ1/mm3,m3+d2d3cnxα−vtα/α2b/3aγmβ1/mm3,m3+d41/2ei−kxα/α+wtα/α.Ifε=−1 and m=D+D2+1, then the explicit solution of equation (2) is(33)u8t,x=±d1cnxα−vtα/α−2b/3aγmβ1/mm3,m3+d2d3cnxα−vtα/α−2b/3aγmβ1/mm3,m3+d41/2ei−kxα/α+wtα/α,wherem32=1/1+m2.


## 3. Conclusion

In the paper, the traveling wave solution of the fractional pGI equation is studied by using the complete discrimination system method. Firstly, we reduce equation (2) to the nonlinear ordinary differential equation by fractional traveling wave transformation. Then, the complete discrimination system is used to construct the classification of all single traveling wave solutions of the fractional pGI equation. Among them, the implicit solutions have not been reported in other literatures. Compared with the existing literature [15–17], our solutions are more abundant, including rational function solutions, Jacobian function solutions, and implicit solutions. It is worth mentioning that the complete discrimination system method is an effective and powerful tool for constructing the traveling wave solution fractional pGI equation. Moreover, the method can be employed to construct the traveling wave solution of general FPDE.



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