Solitary and periodic pattern solutions for time-fractional generalized nonlinear Schrödinger equation

1 Introduction

The Schrödinger equation is the most fundamental equation in quantum mechanics, proposed by Austrian physicist Schrödinger in 1926. Its position in quantum mechanics is equivalent to that of Newton’s equation in classical mechanics. Based on the material wave hypothesis, the Schrödinger equation answers the core question of how wave functions evolve over time and provides methods for finding wave functions in various specific situations.

In recent years, various forms of nonlinear Schrödinger-type equations have emerged in a wide range of research fields. For example, they can be used to describe the nonlinear beam propagation in an optical lattice, propagation of ultrashort optical pulses, propagation of an asymmetric Gaussian beam, propagation of light in photonic and photorefractive crystals with tilted band structures, and so on. One of the fundamental properties of nonlinear Schrödinger-type equations is that they possess optical soliton solutions, which has drawn much attention from researchers. That is because of the fact that the optical soliton could be an ideal subject to transmit optical signals due to the particle-like property of the soliton [1,2]. As a large class of self-similar solutions depicting both bright and dark solitary waves and periodic solutions of nonlinear Schrödinger-type equations, soliton solutions have been obtained and discussed by many researchers [3–6]. Zeng et al. [7] have demonstrated the existence of localized solutions in the nonlinear system of three coupled Schrödinger equations with fractional dispersion, linear coupling, and cubic nonlinearities. Then, they have checked the stability of these localized solutions, i.e., solitons, by using the linear stability analysis method and by propagating the perturbed solutions using the split-step fast-Fourier beam propagation method. Xu and Fu [8] have developed two novel conservative relaxation methods for the space-fractional nonlinear Schrödinger equation. The first type of relaxation scheme has adopted the exponential time difference method in time: the scheme could preserve the energy conservation of the space-fractional nonlinear Schrödinger. The second type of relaxation scheme has developed the integral factor method in time: this scheme could be proved to inherit the mass conservation of space-fractional nonlinear Schrödinger.

Recently, fractional differential equations (FDEs) have received widespread attention due to the exact description of nonlinear phenomena in physics, engineering, fluid flow, and other areas of science [9–11].

The concept of fractional order provides a useful mechanism for dealing with nonlocal phenomena in quantum physics, which can better explain quantum effects in long-term interactions and time-related multiscale quantum effects. In recent years, the framework of the time-fractional Schrödinger equation has been established and used to describe some physical phenomena.

The aim of this study is to establish explicit solutions of the following time-fractional generalized nonlinear Schrödinger equation (GNLSE)

with the initial condition

Here, u ( x , t ) is a complex function, β and γ are the dispersion and nonlinearity management parameters, respectively, V ( x ) stands for the external potential [12,13], D t α ( ⋅ ) is the Jumarie’s modified Riemann–Liouville derivative with order α ( 0 < α ≤ 1 ) [14–16] defined in Section 2. In the following, we shall employ the notations D t α f ( t ) = f ( α ) ( t ) . It is worth mentioning that if the parameters in Eq. (1) are set to some special values, the normalized standard nonlinear Schrödinger equation [6] and the cubic nonlinear Schrödinger equation [17] can be recovered.

On the one hand, fractional order operators have historical dependencies and global correlations. Due to their non local nature, their calculation and storage costs are very expensive. Taking the fractional order derivative of time as an example, let N be the total number of time steps, in the numerical simulation process, not only does it need to store all historical moments, but its storage cost is also very expensive. Long time simulation of fractional order models almost reaches the bottleneck of calculation, How to quickly and effectively approximate fractional derivatives is an urgent problem to be solved.

On the other hand, with the advent of high-performance computers, solving many linear problems has become increasingly easy, but we know that most natural phenomena in the real world are nonlinear phenomena that need to be described by nonlinear equations. But it is very hard to obtain accurate analytical solutions. Even using high-level symbolic computing software Mathematica and Maple to solve analytical solutions for nonlinear problems is very difficult. We know that numerical methods only provide discontinuous points on the solution curve, making it difficult for the results of numerical methods to have a comprehensive understanding and understanding of nonlinear problems. It is always a pleasure to be able to find an analytical solution to a nonlinear problem.

For a better understanding of the exact dynamical behavior of the beam during the propagation process, it is of great importance to establish explicit solutions of the GNLSE. A lot of methods have been applied to obtain solutions of FDEs, such as differential transform method (DTM) [18], finite difference method (FDM) [19], fractional variational iteration method (FVIM) [20], Adomian decomposition method (ADM) [21], and so on.

In this study, we employ the fractional variational iteration method with He’s polynomials (FVIMHPs) [22] proposed recently for solving Eq. (1) with initial condition (2). The FVIMHP provides the semi-analytical solution of the problem in a speedy convergent series, which can make the solution closed and means solving the problem of difficulty in constructing solutions for fractional order equations. In addition, this method achieves algorithmic innovation, and the semianalytical or exact solutions obtained have rarely appeared in previous research, which is also the novelty of the article. Compared with the FVIM, the positive of FVIMHP is introducing He’s homotopy perturbation (He’s polynomials) in the correct functional [23]. Using homotopy perturbation reduces the complicated huge computational work and the continuous application of the integral operator while still maintaining very high accuracy [24]. The modified Riemann–Liouville derivative has the following merits. First, the α th derivative of a constant is zero. Second, compared with the classical Caputo derivative [25], the definition of the fractional derivative is not required to satisfy higher integer-order derivative than α [26]. For its merits, Jumarie’s modified derivative has been successfully applied to the probability calculus [27] and fractional Laplace problems [28]. Also, it provides the Mittag–Leffler function as the solution of a basic linear FDE. Therefore, the FVIMHP is formulated by taking the full advantages of variational iteration method (VIM) [29–31], homotopy perturbation method (HPM) [32,33] and Jumarie’s modified Riemann–Liouville derivative.

This article is organized as follows: in Section 2, some basic definitions of the Jumarie’s modified Riemann–Liouville derivative and the application of FVIMHP to Eq. (1) with initial condition (2) are given briefly. In Section 3, some simpler cases of Eq. (1) are considered. In Section 4, some conclusions are provided.

2 Jumarie’s fractional derivative and the application of FVIMHP

Now, we demonstrate the application of the FVIMHP to Eq. (1) with initial condition (2). First, we establish the correction functional for fractional GNLSE in the following form (1):

where λ ( τ ) is the Lagrange multiplier identified via the variational theory [34]. The subscript n ≥ 0 denotes the n -th approximation, the function u ˜ n is a restricted variation, i.e., δ u ˜ n = 0 . ∣ u ( x , t ) ∣ 2 = u ( x , t ) u ¯ ( x , t ) , and u ¯ ( x , t ) is the conjugate of u ( x , t ) .

The stationary conditions can be obtained as follows:

and

Therefore, the Lagrange multiplier can be identified as λ = i .

Second, using the obtained Lagrange multiplier, we establish the following iteration formula:

which is formulated by combining of FVIM and He’s polynomials. q ∈ [ 0 , 1 ] is an imbedding parameter, and u 0 is an initial approximation of Eq. (1), i.e., u 0 = f ( x ) .

Then, comparing with the coefficients of the same power of q in the both sides of the expression (12), we can obtain u i ( i = 0 , 1 , 2 , … ) . According to the HPM, we have

which is the solution of Eq. (1).

u 0 is the given initial value, u 1 is the first iteration value, u 2 is the second iteration value, u k is the k th iteration value, so the solution of Eq. (1) is a series solution composed of u k ( k = 0 , 1 , 2 , 3 , … ).

3 Results

Solving the fractional GNLSE (1) with the initial condition (2) could allow us to analyze the behavior of the beam during propagation along the fibers. In order to understand the properties of the propagation of the beam, we need first consider some simpler cases.

3.1 Case I

Consider the following fractional GNLSE with power law nonlinearity (focusing effect of the nonlinearity) and zero trapping potential:

(14) i D t α u ( x , t ) + u x x ( x , t ) + 2 ∣ u ( x , t ) ∣ 2 p u ( x , t ) = 0 , p ≥ 1 ,

(15) u ( x , 0 ) = [ 2 ( p + 1 ) sech 2 ( 2 p x ) ] 1 2 p .

The iteration formula could be structured as

∑ n = 0 ∞ q n u n ( x , t ) = u 0 ( x , t ) + q ∑ n = 1 ∞ q n u n ( x , t ) + i Γ ( 1 + α ) ∫ 0 t i ∑ n = 0 ∞ q n D τ α u n ( x , τ ) + ∑ n = 0 ∞ q n [ u n ( x , τ ) ] x x + 2 ∑ n = 0 ∞ q n u n p + 1 ( x , τ ) u ¯ n p ( x , τ ) ( d τ ) α .

From u 0 ( x , t ) = [ 2 ( p + 1 ) sech 2 ( 2 p x ) ] 1 2 p , consequently,

(16) u ( x , t ) = [ 2 ( p + 1 ) sech 2 ( 2 p x ) ] 1 2 p E α ( 4 i t α ) ,

where E α ( z ) = ∑ k = 0 ∞ z k Γ ( 1 + k α ) ( α > 0 ) is the Mittag–Leffler function in one parameter.

It is interesting to point out that when α = 1 , we obtain the single bright soliton solution of the form

(17) u ( x , t ) = [ 2 ( p + 1 ) sech 2 ( 2 p x ) ] 1 2 p e 4 i t ,

immediately upon replacing α by 1 in Eq. (16), which is exactly the same as solution obtained in [17].

Figure 1(a) shows that our solution is consistent with the exact solution. Figure 1(b) shows the tenth-order approximate solution (16) for p when t and α are fixed. It indicates that the solution of Eq. (14) ∣ u ( x , t ) ∣ becomes smaller when the values of p increase. Figure 1(c) shows the approximate solution (16) for α when t and p are fixed. It is evident that the values of ∣ u ( x , t ) ∣ become bigger when the values of α ( ≥ 0.8 ) increase.

Figure 1

(a) The absolute error E between the tenth-order approximation solution and exact solution (17) of Eq. (14) when α = p = 1 , (b) the tenth-order approximate solution ∣ U 10 ( x , t ) ∣ of Eq. (14) for different values of p when t = 0.5 and α = 0.8 , and (c) the tenth-order approximate solution ∣ U 10 ( x , t ) ∣ of Eq. (14) for different values of α when p = 1 and t = 0.5 .

3.2 Case II

Consider another fractional GNLSE with power law nonlinearity (defocusing effect of the nonlinearity) and zero trapping potential

(18) i D t α u ( x , t ) + u x x ( x , t ) − 2 ∣ u ( x , t ) ∣ 2 p u ( x , t ) = 0 , p ≥ 1 ,

under the initial condition

(19) u ( x , 0 ) = [ 2 ( p + 1 ) csch 2 ( 2 p x ) ] 1 2 p .

The iteration formula could be structured as follows:

∑ n = 0 ∞ q n u n ( x , t ) = u 0 ( x , t ) + q ∑ n = 1 ∞ q n u n ( x , t ) + i Γ ( 1 + α ) ∫ 0 t i ∑ n = 0 ∞ q n D τ α u n ( x , τ ) + ∑ n = 0 ∞ q n [ u n ( x , τ ) ] x x − 2 ∑ n = 0 ∞ q n u n p + 1 ( x , τ ) u ¯ n p ( x , τ ) ( d τ ) α .

From u 0 ( x , t ) = [ 2 ( p + 1 ) csch 2 ( 2 p x ) ] 1 2 p , we can derive

q 0 : u 0 ( x , t ) = [ 2 ( p + 1 ) csch 2 ( 2 p x ) ] 1 2 p , q 1 : u 1 ( x , t ) = [ 2 ( p + 1 ) csch 2 ( 2 p x ) ] 1 2 p 4 i t α Γ ( 1 + α ) , q 2 : u 2 ( x , t ) = [ 2 ( p + 1 ) csch 2 ( 2 p x ) ] 1 2 p ( 4 i t α ) 2 Γ ( 1 + 2 α ) , q 3 : u 3 ( x , t ) = [ 2 ( p + 1 ) csch 2 ( 2 p x ) ] 1 2 p ( 4 i t α ) 3 Γ ( 1 + 3 α ) , ⋮

Consequently,

(20) u ( x , t ) = [ 2 ( p + 1 ) csch 2 ( 2 p x ) ] 1 2 p E α ( 4 i t α ) .

When α = 1 , we have

(21) u ( x , t ) = [ 2 ( p + 1 ) csch 2 ( 2 p x ) ] 1 2 p e 4 i t ,

which is an interesting single bright soliton solution (singular soliton) of Eq. (18).

Figure 2(a) shows that our solution is consistent with the exact solution. Figure 2(b) shows the solution (20) for p when t and α are fixed. It indicates that the solution of Eq. (18) ∣ u ( x , t ) ∣ becomes smaller when the values of p increase. Figure 2(c) shows the solution (20) for α when t and p are fixed. It is evident that the values of ∣ u ( x , t ) ∣ become bigger when the values of α ( ≥ 0.8 ) increase.

Figure 2

(a) The absolute error E between the tenth-order approximation solution and exact solution (21) of Eq. (18) when α = p = 1 , (b) the tenth-order approximate solution ∣ U 10 ( x , t ) ∣ of Eq. (18) for different values of p when t = 0.2 and α = 0.8 , and (c) the tenth-order approximate solution ∣ U 10 ( x , t ) ∣ of Eq. (18) for different values of α when p = 4 and t = 0.2 .

3.3 Case III

Consider the following fractional GNLSE with periodic trapping potential

(22) i D t α u ( x , t ) + 1 2 u x x ( x , t ) − ∣ u ( x , t ) ∣ 2 u ( x , t ) − cos 2 ( x ) u ( x , t ) = 0 ,

(23) u ( x , 0 ) = sin ( x ) .

The iteration formula could be structured,

∑ n = 0 ∞ q n u n ( x , t ) = u 0 ( x , t ) + q ∑ n = 1 ∞ q n u n ( x , t ) + i Γ ( 1 + α ) ∫ 0 t i ∑ n = 0 ∞ q n D τ α u n ( x , τ ) + 1 2 ∑ n = 0 ∞ q n [ u n ( x , τ ) ] x x − ∑ n = 0 ∞ q n u n 2 ( x , τ ) u ¯ n ( x , τ ) − ∑ n = 0 ∞ q n cos 2 ( x ) u n ( x , τ ) ( d τ ) α .

From u 0 ( x , t ) = sin ( x ) , we can derive, consequently,

(24) u ( x , t ) = sin ( x ) E α − 3 2 i t α .

The solution (24) is an exact solution of (22), and the E is Mittag–Leffler function in Eq. (16).

It is worth mentioning that when α = 1 ,

(25) u ( x , t ) = sin ( x ) e − 3 2 i t ,

which is exactly the same as solution obtained in the study by Alomari et al. [13].

Figure 3(a) shows that our approximate solution obtained by the method is consistent with the exact solution. Figure 3(b) shows the tenth-order approximate solution (24) for different values of α when t is fixed. It indicates that the periodic pattern solution of Eq. (22) ∣ u ( x , t ) ∣ becomes bigger when the values of α ( ≥ 0.8 ) increase.

Figure 3

(a) The absolute error E between the tenth-order approximation solution and exact solution (25) of Eq. (22) when α = p = 1 and (b) the tenth-order approximate solution ∣ U 10 ( x , t ) ∣ of Eq. (22) for different values of α when t = 1 .

3.4 Case IV

For the fractional GNLSE with periodic trapping potential

(26) i D t α u ( x , t ) + 1 2 u x x ( x , t ) − ∣ u ( x , t ) ∣ 2 u ( x , t ) − sin 2 ( x ) u ( x , t ) = 0 ,

u ( x , 0 ) = i cos ( x ) ,

we can obtain

(27) u ( x , t ) = i cos ( x ) E α − 3 2 i t α .

The solution (27) is an exact solution of Eq. (26), and the E is Mittag–Leffler function in (16).

It is worth mentioning that when α = 1 , we have the following periodic solution

(28) u ( x , t ) = i cos ( x ) e − 3 2 i t ,

which is a new solution of Eq. (26).

Figure 4(a) shows that our solution is consistent with the exact solution. Figure 4(b) indicates that the tenth-order approximate solution (27) for different values of α when t is fixed. It indicates that the periodic pattern solution of Eq. (26) ∣ u ( x , t ) ∣ becomes bigger when the values of α ( ≥ 0.8 ) increase.

Figure 4

(a) the absolute error E between the tenth-order approximation solution and exact solution (28) of Eq. (26) when α = p = 1 and (b) the tenth-order approximate solution ∣ U 10 ( x , t ) ∣ of Eq. (26) for different values of α when t = 1.5 .

Remark 1

According to the given algorithm of FVIMHP in Section 2, one might think the following three important points. First, as an advantage of this method over the ADM, we do not need to do the tedious computations for finding the Adomian polynomials. So, FVIMHP is more direct than ADM. Second, because the FVIMHP does not need to discretize the variables, it is not affected by the round off errors. On the other hand, one is not needed to face with the necessity of large computer memory and time. In addition, we remark that FVIMHP needs less computing time than VIM.

Remark 2

According to the data in Figures 1–4, we can draw the following conclusions:

1. The FVIMHP provides a straightforward and powerful mathematical tool to obtain semi-analytical solutions of the given FDEs. The obtained approximate solutions via FVIMHP are consistent with the given exact solutions.

2. Figures 1(a) and 2(a) indicate that the absolute errors between the approximate solutions and exact solutions of both Eqs. (14) and (18) increase around x = 0 . Figures 3(a) and 4(a) show that the absolute error between the approximate solutions and exact solutions of both Eq. (22) and Eq. (26) is nearly a periodic function of x when the time is fixed.

4 Conclusion

Due to the introduction of fractional order differential operators in the model, the original numerical methods for solving integer-order differential equations cannot be directly applied to the time-fractional order nonlinear Schrödinger equation studied. The FVIMHPs are successfully applied to the fractional GNLSE (1) with initial condition (2). Solitary and periodic pattern solutions of some special cases of Eq. (1) are successfully obtained. Also the relationships between the obtained solutions and the parameters in Eq. (1) are analyzed briefly. Furthermore, the FVIMHP can also be used for other fractional nonlinear systems. However, the convergence of the series has not been proven and can be further studied, which is also an aspect of future research.