Non-classical symmetry and analytic self-similar solutions for a non-homogenous time-fractional vector NLS system

The complex PDEs are a very important and interesting task in nonlinear quantum science. Although there have been extensive studies on the classical complex models, solving the fractional complex models still has a lot of shortcomings, especially for the non-homogenous ones. Therefore, the present study focuses on solving the two-component non-homogenous time-fractional NLS system, our method is to solve a prolonged fractional system derived from the governed model. We first establish non-classical symmetries of this new enlarged system by using the fractional Lie group method. Then, with the help of fractional Erdélyi–Kober operator, we reduce this new system into fractional ODEs, the self-similar solutions are obtained via the power series expansion. The convergence of these solutions are proven as all the variable coefficients are analytic. Finally, we generalize our methods to handle the multi-component case. We conclude that this way may also bring some convenience for solving other complex systems.


Introduction
The vector complex systems have attracted more and more attention in many different fields of nonlinear science during the past few years. To well describe the spins and kinetics of micro-particles, the partial differential equations for these complex systems were set up and widely used in the related ranges of particle physics, quantum mechanics, the condensed matter physics [1][2][3], and many other subjects. One of the most famous models is the nonlinear Schrödinger equations whose general version is governed as iu j t + r j (t, x)u j xx + f j t, x, u 1 , . . . , u m u j = 0 (j = 1, . . . , m). (1.1) tions. There has been abundant research on model (1.1) which explained the kinetics and diffusions of particles in the multi-body quantum regimes. To the best of our knowledge, a lot of soliton waves, breather waves, rogue waves, and periodic waves of Eq. (1.1) were studied by taking advantage of Darboux transformation [4][5][6][7][8][9], inverse scattering method [10,11], Hirota's bilinear transformation [12][13][14], nonlocal symmetry method [15], and many other ways [1-3, 16, 17] in both mathematical and physical points of view. Some mixed type solutions, especially breather-soliton-rogue wave solutions [4,7,9,17], were obtained and used to understand how the quantum waves interact in local excitation patterns.
Recently, models governed by the time-fractional PDEs have been considered in many fields of mechanics and physics [18,19,[29][30][31]. Indeed, the fractional models are more precise than the integer-order ones. For many physical phenomena, different time memories are often represented by different integral kernels of several definitions [20,21], two of the most influence and popularity are Riemann-Liouville type and Caputo type [18][19][20][21][29][30][31] which include the singular kernel, and other definitions may contain the nonsingular kernel. The singular kernel (general kernel), for instant power kernel which was derived by Cauchy integral, describes how the quantity process obeys a singular law by empirical observation in many real problems. The power memory has many good mathematical properties such as self-similarity, semi-group property, Laplace transformation, but the disadvantage is the lack of elaborate statistical tests and empirical support. Thus it should be natural to consider the nonsingular kernel which can show the fading memories with relaxation. The typical type for nonsingular kernel is Caputo-Fabrizio definition [20,21] of exponential memory that may be applied to well understand the stochastic process of empirical distribution, but this expression is more difficult to compute. In short, the singular kernel can more generally characterize the real nonlocal nonlinear phenomenon and is more convenient for calculating, thus it should take precedence to use for solving fractional differential equations. In physical point of view, some micro-structures may often lead to the short time memories effect, the smaller α decides the faster time memory. In addition, the Riemann-Liouville derivative has stronger singularity than Caputo derivative, thus the Riemann-Liouville definition can be often used without initialboundary conditions. Therefore, in the present work we mainly investigate the following non-homogenous fractional NLS system with Riemann-Liouville time derivatives: This fractional system more precisely characterizes the Bose-Einstein concentration and phase transition behaviors of critical states than the integer one in the two-body quantum regimes, where the fractional derivatives ∂ α u ∂t α , ∂ α v ∂t α describe two wave functions with nonlocal time memories, and r(t, x), s(t, x), f (t, x, |u|, |v|), g(t, x, |u|, |v|) are variable coefficients as (1.1).
However, solving fractional system (1.2) is really a new and difficult work. On one hand, since integrability of the fractional models is much poorer than that of the classical ones, the compound function solutions of (1.2), for typical traveling wave solutions, were hardly obtained by adopting some direct methods. On the other hand, there have been abundant studies on Lie symmetries, conservation laws, and exact explicit solutions for many integer and fractional real PDEs [22][23][24][25][26][27][28][32][33][34][35][36][37][38][39][40][41][42][43]. However, few symmetries of the time-fractional complex system have been discussed until now, even non-homogenous ones. For the classical n-component complex PDE systems, the common method is to split the real and imaginary parts of two complex variables u, v and compute the symmetries of 2n equations with 2n variable coefficients r, s, f , g, this may cause some difficulties. To solve this problem in a concise way, we introduce the complex conjugations u * , v * and regard functions f , g as two new functions. Here, in order to close the system, we also need to relate f , can enlarge the vector fNLS model to a new closed fPDE system and only consider solving the new prolonged system. It is novel to construct the symmetries of the prolonged fractional equations since the non-classical symmetries of prolonged system always contain the classical symmetries of the governed model. We also verify that our results can be extended to the more general N-component case by introducing f i = f i (t, x, |u 1 |, |u m |), (i = 1, . . . , N )) and differential system u j f j u ju j * f u j * = 0, (j = 1, . . . , N). The rest of the paper is organized as follows. The non-classical symmetries of prolonged complex system are discussed in Sect. 2. Then, in Sect. 3, this system is reduced by virtue of the Eydélyi-Kober fractional differential operator, and self-similar solutions are acquired by the power expanding method in the de-focused case. We also verify the convergence of solutions in Sect. 4 by using induction as all the coefficients are analytic. Finally, our results are extended to the multi-component case. The concluding remark of our work is put in the last section.

Non-classical symmetry for two-component fractional NLS system
This section considers the non-classical symmetry of system (1.2). By introducing two new conjugate variables u * , v * , we consider the following enlarged complex system: Here, we regard f , g as two new functions. Under the continuous transformation group with infinitesimal generators ξ , τ , , * , , * , F, G, the vector field of the generators of Lie group is given by and the α, 2-order prolonged vector field is shown as where τ , ξ , F, G are real functions and , are complex ones. Applying the Lie symmetry method to system (1.2) yields the following results.

Notation
In the following proof, we denote by C n α a combination number where C n Proof By adopting the fractional Lie group method, the invariance of system (2.1) is determined by the following linear equations: with the prolonged generators Substituting (2.7) into (2.6) with the help of prolonged system (2.1), after equaling the coefficients of all derivatives of u, v, we have admitted conditions as follows: Solving the linear PDEs (2.8) one by one leads to the desired results. The next result shows the self-similar reduction.

Self-similar solution for two-component fractional NLS system
Let us consider the scaling action V = V 1 + σ V 2 , where the parameter is chosen as σ = -c 2 c 1 . In this section, we search for the self-similar solutions of system (1.2).
Theorem 2 When we take ξ = x 2 1+σ t -α , under the scaling action V , system (2.1) can be reduced to the following fractional ODEs: Here, the Erdélyi-Kober fractional differential operator is defined as Proof Herein we use the invariance to construct self-similar solutions. First, solving the following characteristic gives rise to Then, by using the chain rule, the prolonged parts of system (2.1) become solving these four linear PDEs yields On the other hand, from the definition of fractional Erdélyi-Kober differential operator, we obtain the fractional derivatives as follows: In the same way we have where P (σ -1)α- 2 4 ,α In addition, other terms of system (2.1) become and  (v k v * m-l-ku k u * m-l-k ) + (a 0 u n + a n u 0 )(|v 0 | 2 -|u 0 | 2 ) ( 2 1+σ ) 2 ( 1-σ 2 + n)(n + 1)r 0 1+σ t (σ -4n-1)α- 2 4 , and v(t, x) where a n , b n , r n , s n are expanding coefficients of A, B, R = ξ R, S = ξ S, and n = are the parameters.
Proof Under the analytic assumptions, according to (3.2) and ( (3.10) We suppose that the solutions of (3.10) are formed as follows: and 11) where u n , v n are unknown expanding coefficients. Substituting (3.11) into the first term of (3.10)to simplify the fractional terms of (3.1), we have P (σ -1)α- 2 4 ,α 1 α u n ξ n , (3.12) and where we use the integral .

Convergence analysis for self-similar solution
We prove the convergence of solutions (3.9) in this section. Proof The key scheme of the proof is to construct the majorant series by using the induction.
By virtue of the analytic assumption of R, S, A, B, now we select p n+1,R , p n+1,I , q n+1,R , q n+1,I as the right-hand side of (4.10) and notice that all Finally, we can set up four majorant functions as follows: (4.12) From (4.2)-(4.8) we have bounded all the first three terms of (4.12). On the interval 0 < |ξ | < 1, the series ∞ n=1 ξ n+1 converges to ξ 2 1-ξ , this ends the proof.

Extension to m-component case
In this section, we verify that the above results of system (1.2) can also be extended to the m-component fractional NLS model i ∂ α u j ∂t α + r j (t, x)u j xx + f j t, x, u 1 , . . . , u m u j = 0 (j = 1, . . . , m), (5.1) which describes the kinetics of multi-body quantums with time-memories and nonlinear interactions. By introducing |u j | = √ u j u j * , we discuss the following prolonged system: iD α t u j + r j (t, x)u j xx + f j t, x, u 1 , . . . , u m u j = 0, where the diffusion coefficients solve τ r j t + ξ r j x + (ατ t -2ξ x )r j = 0 (j = 1, . . . , m).