Exact solutions and dynamic properties of perturbed nonlinear schrödinger equation arised from nano optical fibers

The main idea of this paper is to investigate the exact solutions and dynamic properties of a space-time fractional perturbed nonlinear schrödinger equation involving Kerr law nonlinearity with conformal fractional derivative. Firstly, by the complex fractional traveling wave transformation, the traveling wave system of the original equation is obtained, then a conserved quantity, namely the Hamiltonian is constructed, and the qualitative analysis of this system is conducted via this quantity by classifying the equilibrium points. Moreover, the prior estimate of the existence of the soliton and periodic solution is established via the bifurcation method. Furthermore, all exact traveling wave solutions are constructed to illustrate our results explicitly by the complete discrimination system for polynomial method.


Introduction
For hundred of years, partial differential equation plays a vital role in many fields of science, and constructing exact solution to it could help us gain a deep insight into the phenomena revealed. However, traditional integer-order equation sometimes could not meet the requirement of modeling some special problems such as anomalous diffusion [1], thus the fractional calculus is proposed to handle this. Fractional calculus has many definitions, for example, the traditional Riemann-Liouville (RL) definition [2], modified RL definition [3] and conformal definition [4]. However, the classical RL definition is very complex to apply, and the modified RL definition has already been proved wrong [5][6], thus choosing a proper fractional definition is an important and difficult task.
In this paper, we consider the following space-time fractional perturbed nonlinear schrödinger equation with conformal fractional derivative, where α is the corresponding fractional order, q(x, t) is the complex valued function defining wave profile in optical fibers, a and b represent the group velocity dispersion and nonlinear term, respectively, σ is the intermodal dispersion, λ is the self-steepeninng perturbation term and γ is the nonlinear dispersion coefficient. This equation has a broad applications in modeling light waves in nano optical fibers. Especially, when the external electric field exists, this equation can be used to solve non-harmonic motion of electrons bound in molecules [7], so constructing exact solutions to this equation is of great significance. In [8], some optical solitons and singular periodic wave solutions are constructed by the extended Sinh-Gordon equation expansion method, and W-shaped solitons are shown by Al-Ghafri and his colleagues [9]. In this paper, we conduct qualitative and quantitative analysis to (1) by the complete discrimination system for polynomial method (CDSPM). The topological structure of this equation is shown and the prior estimate of the existence of soliton and periodic solution is also presented. Moreover, to verify our conclusion explicitly, all exact traveling wave solutions, namely the classification of traveling wave solutions is gotten. All conditions of parameters are discussed, thus this paper contains all results of traveling wave solutions in the existing literatures, and some new solutions are obtained. The CDSPM is proposed by Liu [10][11][12], and has been successfully applied to a series of integerorder [13][14] and fractional order equations [15][16][17][18][19]. Then Kai found that this method could also be used to conduct qualitative analysis [20], and combining with the bifurcation method, we can even establish the existence of the soliton and periodic solution [21]. The construction of this paper is as follows. The corresponding traveling wave system is given in Sec. 2 and qualitative analysis is conducted. Moreover, the existence of the soliton and periodic solution is also established in this section. To verify our conclusion explicitly, all single traveling wave solutions are constructed in Sec. 3 and concrete examples under concrete parameters are also shown to ensure the existence of each solution. In the final, a brief discussion is given.
2 Dynamic properties of Eq. (1) where ξ = x α α − vt α α [18], (1) becomes for the real part, and for the imaginary part. Substituting (3) into (4) yields where a 2 and C is a constant of integration. By setting u 2 = V , we have Multiplying (6) with V and integrating it once, we have where a 4 = 2A3 3 , a 3 = A 2 , a 2 = 2A 1 , a 1 = 4A 0 and a 0 is a constant of integration. (6) is equivalent to the following dynamic system and thus the corresponding Hamiltonian is given by Now let us show that the Hamiltonian (9) is a conserved quantity. Taking derivative of right side of (9) with respect to ξ, we have which just proved our conclusion, and we can also conclude that the global phase portrait to system (8) is just the contour line of the Hamiltonian (9). In the following, we shall conduct qualitative analysis through this quantity by introducing the complete discrimination system.
From the Hamiltonian (9), we can see that the derivative of the potential energy is given by where b 2 = 3a3 4a4 , b 1 = a2 2a4 and b 0 = a1 4a4 . By introducing following discrimination system we can see that four cases need to be discussed here. Case 1. D < 0, ∆ = 0, we can get  From Fig. 1b, we can see that trajectory I is a closed orbit with a center inside, which just indicates the existence of the periodic solution, and trajectory II is a homoclinic orbit, which indicates the existence of the bell-shaped soliton solution [21].
Case 2. D = 0, ∆ = 0, we have There is only one equilibrium point (s, 0) here. It is a cuspidal point when a 4 > 0 and a center when a 4 < 0. For example, when a 3 = a 2 = a 1 = 0, we have s = 0, and the corresponding global phase portraits can be seen in Fig. 2 when a 4 = ±1. From Fig. 2b, we can also conclude that the original equation has periodic solution.
Case 3. ∆ < 0, we have There is also only one equilibrium point (s, 0) in this case. It is a saddle point when a 4 > 0 and a center when a 4 < 0. So this case is very similar to Case 2. For example, when a 3 = a 1 = 0 and a 2 = 2, we have s = −1, and the global phase portrait is given in Fig. 3 when a 4 = ±1. Case 4. D < 0, ∆ > 0, we have This case is rather interesting due to that there are three equilibrium points  Fig. 4.
For Fig. 4a, we can see that trajectory I is a closed orbit with a center inside, which indicates the existence of the periodic solution, and trajectory II and III are two heteroclinic orbits, which indicates the existence of the kink and anti-kink solitary wave solution, respectively. Fig. 4b is a "figure eight loop" with trajectoris I and II are two closed orbits with a center inside, which indicate the existence of the periodic solution, and trajectories III and IV are two homoclinic orbits, which means the corresponding equation has bright and dark bell-shaped soliton solution.
Now we have showed the topological structure of the system (8) and established the existence of the soliton and the periodic solution. In order to verify the conclusion explicitly, we construct the classification of traveling wave solutions to (7) by the CDSPM.

Traveling wave solutions to (7)
In this section, we construct all traveling wave solutions, namely the classification of traveling wave solutions to (7). We only focus on the condition of a 4 > 0, and a 4 < 0 could also be treated similarly.
By taking the following transformation ϕ = (a 4 ) 1/4 (u + a 3 4a 4 ), ξ 1 = (a 4 ) 1/4 ξ, where p = a2 √ a4 , q = ( − a1a3 4a4 + a 0 . First, we need to introducing the following complete discrimination system The complete discrimination system (12) given in Sec. 2 is the third order form, and here is the fourth order. By discussion the relation between the parameters and the coefficients, we shall see that every condition of parameters is discussed, thus what we have gotten is the classification of traveling wave solutions.
which leads to For example, when p = q = r = 0, we have Case 3 When D 2 > 0, D 3 = D 4 = 0 and E 2 > 0, F (ϕ) has two double distinct real roots, then we have which yields If ϕ > s or ϕ < l, we can get and when l < ϕ < s, we have (29) is a solitary wave solution. (21) and (29) has verified the conclusion given in Sec. 2 that when a 4 > 0, (7) has periodic and soliton solution. This shows that the qualitative results obtained are truly correct.

Conclusion
In this paper, we consider a space-time fractional perturbed nonlinear schrödinger equation arised from nano optical fibers. By taking the complex fractional traveling wave transformation, the traveling wave system of the original equation is gotten, then the corresponding Hamiltonian is constructed, and the qualitative analysis is conduct by introducing the complete discrimination system. The topological structure is given and the existence of the soliton and periodic solution is established via the bifurcation method. To verify our conclusion explicitly, every kind of traveling wave solutions are constructed by the CDSPM, and some of them are new. The results show the powerfulness and the efficient of the method adopted in the paper.