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OPEN The impact of memory efect on space fractional strong couplers with tunable decay behavior and its numerical simulation Ahmed S. Hendy 1,2, Mahmoud A. Zaky 3*, Ramy M. Hafez 4 & Rob H. De Staelen 5,6

The nontrivial behavior of packets in the space fractional coupled nonlinear Schrödinger equation has received considerable theoretical attention. The difculty comes from the fact that the Riesz fractional derivative is inherently a prehistorical . In contrast, nonlinear Schrödinger equation with both time and space nonlocal operators, which is the cornerstone in the modeling of a new type of fractional quantum couplers, is still in high demand of attention. This paper is devoted to numerically study the propagation of solitons through a new type of quantum couplers which can be called time-space fractional quantum couplers. The numerical methodology is based on the fnite- diference/Galerkin Legendre spectral method with an easy to implement numerical algorithm. The time-fractional derivative is considered to describe the decay behavior and the nonlocal memory of the model. We conduct numerical simulations to observe the performance of the tunable decay and the sharpness behavior of the time-space fractional strongly coupled nonlinear Schrödinger model as well as the performance of the numerical algorithm. Numerical simulations show that the time and space fractional-order operators control the decay behavior or the memory and the sharpness of the interface and undergo a seamless transition of the fractional-order parameters.

Recently, optical solitons have arisen in various optical systems, which have many ­applications1–3. Tere exist dif- ferent classifcations of optical solitons. Temporal solitons and spatial solitons are the two main classifcations. Te combination of these two classifcations forms spatiotemporal solitons. Tis kind of solitons, which are named bullets, can be formed in the support of nonlinear efect in which a compensation of light beams spatial difraction (broadening) and pulses temporal dispersion (spreading) occurs­ 1,3,4. Information technology, like the internet and other telecommunication networks, is based on optical fber technology. Tis technology is used widely in biomedical and biological studies. In the fabrication process of conventional optical fbers, it is noticed that some optical properties, apart from the transmission of light, can’t be easily modifed and preserved. Routing, switching, and bufering of information are some of these optical properties. Because of that, the eforts of researchers in recent years are devoted to using dual-core fbers to develop and enhance fber optic sensing techniques­ 5,6. Te ability of the new dual-core fbers to perform the same functions more precisely is proved in­ 7 by applying a minuscule amount of mechanical pressure. In other words, an improvement is reported in opti- cal fber-based technology by the mechanism of two cores fber fabrication which is based on moving the cores close enough to each other. More recently, a nanomechanical optical fber fabrication and demonstration are reported­ 7. Te main feature of that optical fber is its core which has a great role in controlling the nanometer-scale mechanical movements side by side to the transmission of light. Te methodology of the fabricated fbers which have two movable cores

1Department of Computational Mathematics and Computer , Institute of Natural and Mathematics, Ural Federal University, 19 Mira St., Yekaterinburg, Russia 620002. 2Department of Mathematics, Faculty of Science, Benha University, Benha 13511, Egypt. 3Department of Applied Mathematics, Division, National Research Centre, Dokki, Cairo 12622, Egypt. 4Faculty of Education, Matrouh University, Matrouh, Egypt. 5Dean’s Ofce of the Faculty of Medicine and Health Sciences, Ghent University, C. Heymanslaan 10, Ghent 9000, Belgium. 6Ghent University Hospital, C. Heymanslaan 10, Ghent 9000, Belgium. *email: ma.zaky@ yahoo.com

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is based on making their cores more close to each other to have the opportunity to act as a directional coupler. Armed by that, the light can make a jump between the two cores which are separated only by a few nanometers. Some recent applications are based on that new technology. An example of these applications is optical bufering which can enhance processing, routing, and switching properties over long distances. Owing to potential applica- tions in optical communication, propagation of solitons through optical fbers in most optical communication systems became a wide area of research. Te integrable nonlinear Schrödinger’s equation (NLSE)8,9 is used to model these dynamics. A more generalized form­ 10, known as the generalized nonlinear Schrödinger’s equation (GNLSE), is given by

1 2 iut uxx F u u 0, (1) + 2 + | | =  where F is a real-valued algebraic function which can be considered as the of the fbers. Earlier ­work10 extended the GNLSE to the couplers case. Fast switching and signal coupling in optical communication links can be allowed by optical nonlinear couplers. For twin-core couplers, at relatively high feld intensities is described by coupled nonlinear equations. In the dimensionless form, they are:

1 2 iut uxx F u u Kν, (2) + 2 + | | = 

1 2 iνt νxx F ν ν Ku. (3) + 2 + | | =  Te coupling coefcient between the cores of the fber is represented by the constant K . Te system (2) and (3) has been derived and studied before in the context of Kerr law nonlinearity­ 11. Te dimensionless forms of the optical felds in the respective cores of the optical fbers are given by u and ν . Tis system of equations is known as the generalized vector NLSE. Intensity dependent switches and devices for separating a compressed from its broad ‘pedestal’ are examples of applications of (2) and (3). Its integrals of motion can be respectively given by energy (E), linear (M) and the Hamiltonian (H) namely:

∞ 2 2 E ( u ν )dx, (4) = | | +| | −∞

i ∞ M (uux∗ u∗ux) (ννx∗ ν∗νx) dx, (5) = 2 − + − −∞  

∞ 1 2 2 2 2 H ux νx F u F ν k uν∗ νu∗ dx, (6) =  2 | | +| | − | | − | | − +  −∞         I where we defne F(I) F (ζ)dζ, and the intensity I is given by I u 2 or I ν 2 depending on the core. Te = 0 =| | =| | energy E is commonly known as the wave energy and in the context of fber it is known as the wave power. Te theory which used to discuss quantum phenomena in fractal environments is called fractional . Te similarity between the classical difusion equation and the Schrödinger equation motivated the generalization of the nonlinear Schrödinger equation in the light of non-Brownian motion in a path integral formulation. Tis generalization leads to the space-fractional, the time-fractional, and the time-space-fractional Schrödinger equations (FSEs). Te Riesz space FSE has been introduced by Laskin in quantum physics by replacing the Brownian paths in the Feynman path integrals by Lévy ­fights12,13. Similar to the conventional Schrödinger equation, the Riesz space FSE satisfes the Markovian evolution law. Stickler­ 14 discussed the Lévy crystal in a condensed matter environment as a possible realization of the space-fractional quantum mechanics by introducing a tight binding infnite range chain. ­Longhi15 discussed an optical realization of the space FSE based on transverse light dynamics in a cavity by exploiting the properties. Other physical applications of the Riesz space FSE have been discussed by Guo and Xu­ 16 and its solution for a free particle and an infnite square potential well have also been introduced. Te existence and uniqueness of the solution to the space FSE have been investigated by Guo et al.17 using the energy method. Moreover, the existence and uniqueness of the solution to systems of the space FSEs have been proved by Hu et al.18 using the Faedo-Galerkin method. Cho et al.19 studied the low regularity well-posedness of the space FSE with cubic nonlinearity in periodic and non periodic settings. Following Laskin and similar to deriving the time-fractional difusion equation by consider- ing non-Markovian ­evolution20, Naber­ 21 used the Caputo temporal fractional ­derivative22 as a generalization of the integer-order derivative in the conventional Schrödinger equation to study non-Markovian evolution in quantum mechanics and constructed the temporal FSE. More recently, Dong and Xu­ 23, and Wang and ­Xu24 combined Laskin’s work with Naber’s work to construct space-time FSEs. A detailed derivation and numerical simulation of coupled system of nonlinear Schrödinger equations for pulses of polarized electromagnetic in cylindrical fbers was investigated­ 25. Ghalandari and Solaimani­ 26 investigated the numerical treatment of the fractional Young double-slit experiment with incident Gaussian wavepackets using a split step Fourier method. Liangwei Zeng and Jianhua ­Zeng27 proposed a coupled system of space FSEs considering two arrays of quantum waveguides. Tey considered the following system with the u, ν:

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α 2 2 ∂u 2 ∂ 2 i Dα u G1F u u Kν, ∂τ = − ∂x2  − | | − α   (7) 2 2 ∂ν 2 ∂ 2 i Dα ν G2F ν ν Ku, ∂τ = − ∂x2  − | | −   where represents the , the diferentiation order parameter α represents the Lévy index, τ is the time, and K is the linear coupling parameter. Te parameter Dα 1/m is constant with m denoting the mass of = the atom. When F (S) S , then the nonlinear term is of Kerr style and G1,2 > 0 are the Kerr coefcients. Tese = coefcients can make a characterization of atoms collisions strengths in Bose–Einstein condensates. A kind of α α normalization can be applied on (7) by defining new variables t τ/τ0, ψ √h− u, φ √h− ν, α 1 α = = = k K/(Dα ), g1,2 G1,2/Dα, and τ0 − /(Dα). The fractional Laplacian is defined as = α/=2 = 2 α/2 ∂2 ∂x2 . Accordingly, the system (7) should have the following dimensionless form: −∇ = −   α ∂ψ 2 2 2 i ψ g1 F ψ ψ kφ, ∂t = −∇ − | | −  α  (8) ∂φ 2 2 2 i φ g2 F φ φ kψ. ∂t = −∇ − | | −   Based on (8), two arrays of quantum waveguides composition represent a new kind of space-fractional quantum couplers was considered in­ 27. It is called space-fractional quantum couplers­ 27. Tis can be understood in the sense of two branches polariton condensates in solid state physics­ 28. Te wave packets in such nonlinear fractional systems are modeled by the coupled system of nonlinear fractional Schrödinger equations with linear coupling. Te new version of the space-time fractional Schrödinger equation derived by Laskin­ 29 contains two scale dimensional parameters. One of them is a time-fractional generalization of the famous Planck’s constant. Te other one can be explained as a time-fractional generalization of the scale parameter emerging in fractional quantum mechanics. Ten the dimensionless physical model that can efectively describe the time-fractional quantum coupling is based on a set of strongly NLFSEs as follows,

∂β ψ α/2 2 2 i β ε( �) ψ k1 ψ (k1 2k2) φ ψ ξφ 0, x �, t J, ∂t − − + | | + + | | + = ∈ ∈  � �  β  i ∂ φ ε( �)α/2φ k φ 2 (k 2k ) ψ 2 φ ξψ 0, x �, t J,  ∂tβ 1 1 2  − − + | | + + | | + = ∈ ∈ (9)  � � ψ(x,0) ψ0, φ(x,0) φ0, x �, = = ∈    ψ(x, t) φ(x, t) 0, x ∂�, t J,  = = ∈ ∈  where � (a, b) R and J (0, T R are the time and the space domains, ∂� is the boundary of , the = ⊂ = ]⊂ parameter ε>0 is the group velocity dispersion, γ denotes the normalized birefringence constant, ψ and φ are complex functions defned in J , ψ0(x) and φ0(x) are known sufciently smooth functions. Of particular × concern for the present work is the linear coupling parameter ξ . It accounts for efects that result from the twisting and elliptic deformation of the fber. Te term proportional to k1 explains the self-focusing of a signal for pulses 2 in birefringent media whereas k1 β k2 represents the cross phase modulation describing the integrability of the ∂ +(0 <β<1) ( �)α/2 nonlinear system. Te operator ∂tβ is the Caputo fractionalα derivative, and is the fractional ∂ − 30 order Laplacian operator of order 1 <α 2 defned in Riesz form α (1 <α<2) . We refer to seminal ­works ≤ ∂ x for more details on fractional order diferential operators. We here| numerically| solve model (9) and provide an easy to implement algorithm in the next section. In this paper, we numerically investigate the nontrivial behavior of wave packets in the time-space fractional model (9) in one space dimension. In this model, we choose the form of the fractional Laplacian to be the Riesz fractional operator. Te time-fractional derivative is considered to describe the decay behavior and the non-local memory of the model. Te numerical methodology is based on the fnite diference/Galerkin–Legendre spectral method with an easy to implement numerical algorithm. We conduct numerical simulations to demonstrate the performance of the tunable decay and the sharpness behavior of the time-space fractional strongly coupled nonlinear Schrödinger model as well as the the performance of the numerical algorithm. Numerical simula- tions show that the time and space fractional order operators control the decay behavior or the memory and the sharpness of the interface and undergo a seamless transition of the fractional order parameters. Of particular interest for this work also is the linear coupling parameter ξ . We test the efect of this parameter on the collision of solitary waves. The spectral scheme Here, the discretization of problem (9) is done by using the high order L2-1σ approximation diference formula for the Caputo time fractional operator next to the spectral Legendre–Galerkin scheme for the Riesz spatial- fractional operator. Te time and space fractional Schrödinger equalion and its coupled system are fully analyzed numerically ­in31,32. Te proposed numerical approaches there are designed by the use of L1 diference scheme for temporal approximations and Galerkin spectral scheme for spatial fractional order operators. Alternatively, Fourier spectral method can be used efectively for Riesz space fractional partial diferential equations. For example, a fast and accurate method for numerical solutions of space fractional reaction-difusion equations is proposed in­ 33 based on an exponential integrator scheme in time and the Fourier spectral method in space. By the

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Fourier spectral techniques and advance the resulting equation in time with both Strang splitting and exponential time-diferencing methods, the dynamics of the time-dependent Riesz space fractional nonlinear Schrödinger equation in the presence of the harmonic potential have been considered in­ 34. A detailed implementation of the constructed high accuracy algorithm for (9) will be given below.

Formalism. We partition the temporal domain J by ts sτ, s 0, 1, ..., M where τ T/M . Denote = s=σ s σ = ts σ (s σ)τ σts 1 (1 σ)ts, for s 0, 1, ..., M 1 . Let � + � + ( ) �( , ts σ ). + = + = + + − = − = · = · + β Defnition 1 Let 0 <β<1 and σ 1 . Defne = − 2 1 β (β σ) σ , s 0, a , − = s = (s σ)1 β (s σ 1)1 β , s 1, (10) + − − − + − ≥

(β,σ) 1 2 β 2 β 1 1 β 1 β b (σ s) − (s σ 1) − (s σ) − (s σ 1) − , s 1, s = 2 β + − − + − 2 + + − + ≥ (11) −   and (β,σ) a0 , s j 0,  (β,σ) (β,σ) = = (j,β,σ) a0 b1 , s 0, j 1, Cs  (β,σ) + (β,σ) (β,σ) = ≥ (12) =  as bs 1 bs ,1 s j 1, + + − ≤ ≤ − a(β,σ) b(β,σ),1s j.  j − j ≤ =  35 Lemma 2.1 (Alikhanov diference ­formula ) Te high order Alikhanov L2-1σ diference formula under the assump- 3 tion �(t) C 0, tj 1 , 0 j M 1 , formulated as ∈ [ + ] ≤ ≤ − j τ β β j σ − (j,β,σ) r O 3 β 0Dτ � + Cj r δt � (τ − ),0<β<1, (13) = Ŵ(2 β) − + r 0 − = r r 1 r where δt � � � , can be rewritten as = + − β j β j σ τ − (j,β,σ) r 3 β 0D � + dr � O (τ − ), (14) τ = Ŵ(2 β) + r 0 − = (0,β,σ) (0,β,σ) where d d σ 1 β j 0 , and j 1, 1 =− 0 = − ∀ = ∀ ≥ (j,β,σ) Cj , s 0, (j,β,σ) −(j,β,σ) (j,β,σ) = ds  C C ,1 s j,  j s 1 j s (15) =  (−j,β+,σ) − − ≤ ≤ C , s j 1. 0 = +  Defnition 2 Let j Z 0,M 1 , the Alikhanov L2-1σ diference formula at the node tj σ is defned as ∈ [ − ] + j 1 β + β j σ τ − (j,β,σ) r 0D � + dr � ,0<β<1. (16) τ = Ŵ(2 β) r 0 − = Te next identity holds directly by Taylor’s theorem.

Lemma 2.2 Te following identity holds: O 2 �( , tj σ ) σ�( , tj 1) (1 σ)�( , tj) (τ ). (17) · + = · + + − · + Starting from the L2-1σ formula (16) for the discretization of the time Caputo fractional derivative, we obtain that

β j σ α/2 j σ j σ 2 j σ 2 j σ j σ i 0Dτ ψ ε( �) ψ k1 ψ (k1 2k2) φ ψ γψ ξφ 0, + − − + + + + + + + + + + =  � � � � � �  � � � �  β j σ α/2 j σ j σ 2 j σ 2 j σ j σ (18)  i 0Dτ φ ε( �) φ k1 φ (k1 2k2) ψ φ γφ ξψ 0, + − − + + + + + + + + + + = � � � � � �  � � � �  Making use of Lemma 2.1 and Lemma 2.2, this scheme is of second order accuracy in time. Let us introduce the following parameters 1 (j,β,σ) − (β,σ) (j,β,σ) (β,σ) dj 1 (j,β,σ) ξj di ξ  +  , d ,0 i j. j = τ β Ŵ(2 β) ˜i = τ β Ŵ(2 β) ≤ ≤  −  −

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Te semi-scheme (18) has the following equivalent form: j j 1 (β,σ,γ ) α/2 j 1 (β,σ,γ ) α/2 j (j,β,σ ) q i ψ + εσξj ( �) ψ + ε(1 σ)ξj ( �) ψ dq ψ  − − = − − − q 0  �=  (β,σ,γ ) j 1 2 j 1 2 j 1 (β,σ,γ ) j 1  σξj k1 ψ + (k1 2k2) φ + ψ + σξj ξφ +  − � + + � −  (β,σ�,γ ) � j 2 � �j 2 j (β,σ,γ ) j  (1 σ)ξ � k1�ψ (k1 2�k2) φ� ψ (1 σ)ξ ξφ ,  − − j + + − − j  � � � � � �  � � � �  j (19)  j 1 (β,σ,γ ) α/2 j 1 (β,σ,γ ) α/2 j (j,β,σ ) q i φ + εσξj ( �) φ + ε(1 σ)ξj ( �) φ dq φ − − = − − − q 0  �=  (β,σ,γ ) j 1 2 j 1 2 j 1 (β,σ,γ ) j 1  σξ k1 φ + (k1 2k2) ψ + φ + σξ ξψ +  − j + + − j  (�β,σ�,γ ) � 2 � � �2 (β,σ,γ )  (1 σ)ξ k φj (k 2k ) ψj φj (1 σ)ξ ξψj  j � 1� 1 � 2 � j  − − � + + � − −  � � � �  � � � �  j 1 j 1 0 And so, the full discrete Alikhanov L2-1σ Galerkin spectral scheme for (18) is to get ψN+ , φN+ VN , 0 ∈ j 0, ν V such that ≥ ∀ ∈ N j j 1 (β,σ,γ ) α/2 j 1 (β,σ,γ ) α/2 j (j,β,σ ) q i ψ + , v εσξ ( �) ψ + , v ε(1 σ)ξ ( �) ψ dq ψ , v  N − j − N = − j − N − N � � � � q 0 � �  2 2 �=  (β,σ,γ ) j 1 j 1 j 1 (β,σ,γ ) j 1  σξj IN k1 ψN+ (k1 2k2) φN+ ψ + , v σξj ξ φN+ , v  − � � � � + + � � � � − � �  � � 2 � � 2  (β,σ,γ ) � � j � � j j (β,σ,γ ) j  (1 σ)ξj IN k1 ψN (k1 2k2) φN ψN , v (1 σ)ξj ξ φN , v ,  − − � � � � + + � � � � − − � �  � � � �  � � � �  j  j 1 (β,σ,γ ) α/2 j 1 (β,σ,γ ) α/2 j (j,β,σ ) q  i φN+ , v εσξj ( �) φN+ , v ε(1 σ)ξj ( �) φN dq φN , v � � − − � � = − − − q 0 �= � �  (β,σ,γ ) j 1 2 j 1 2 (β,σ,γ ) j 1  σξ I k φ + (k 2k ) ψ + φj 1, v σξ ξ ψ + , v  j N 1 N 1 2 N + j N  − � � � � + + � � � � − � �  � � 2 � � 2  (β,σ,γ ) � � j � � j j (β,σ,γ ) j  (1 σ)ξj IN k1 φN (k1 2k2) ψN φN , v (1 σ)ξj ξ ψN , v ,  − − � � � � + + � � � � − − � �  � � � �  � � � �  0 0  ψN PN ψ0, φN PN φ0,  = =  (20) where PN is the projection operator.

Algorithmic implementation. Via the hypergeometric function, Jacobi polynomials can be ­represented36, for α, β> 1 and x ( 1, 1): − ∈ − α,β (α 1)i 1 x J (x) + 2F1 i, α β i 1 α 1 − , x ( 1, 1), i N, i = i − + + + ; + ; 2  ∈ − ∈ (21) ! where the notation ( )i represents the symbol of Pochhammer. Armed by (20), we get the equivalent three-term · recurrence relation

Jα,β (x) 1, 0 = 1 1 Jα,β (x) (α β 2)x (α β), 1 = 2 + + + 2 − (22) α,β ( ) α,β α,β α,β ( ) α,β α,β ( ) Ji 1 x ai x bˆi Ji x ci Ji 1 x , i 1, + = ˆ −  −ˆ − ≥ where (β 1 2i α)(β 2 2i α) aα,β + + + + + + , ˆi = (i 1)2(1 i β α) + + + + 2 2 α,β (2i β α 1)(α β ) b + + + − , (23) ˆi =−(1 i)2(i β α 1)(β 2i α ) + + + + + + + (2i β α 2)(i α)(i β) cα,β + + + + + . ˆi = (1 i)(i β α 1)(2i β α) + + + + + + Te Legendre polynomial Li(x) is a special case of the Jacobi polynomial, namely

0,0 x 1 L (x) J (x) 2F1 i,1 i 1 − . i = i = − + ; ;− 2  (24)

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Figure 1. Plots of model (36) for β = 0.99 and α = 1.99.

The weight function that ensures the orthogonality of Jacobi polynomials is given by ωα,β (x) (1 x)β (1 x)α , namely, = + − 1 α,β α,β α,β α,β Ji (x)Jj (x)ω (x)dx γi δij, (25) 1 = − where δij is the Kronecker Delta symbol, and (α β 1) α,β 2 + + Ŵ(1 i β)Ŵ(1 i α) γ + + + + . (26) i = i (α 2i β 1)Ŵ(α β i 1) ! + + + + + +

Lemma 2.3 (see for ­example37) For α>0 , one has

α Ŵ(q 1) α α, α 1Dx Lq(x) + (1 x)− Jq − (x), x 1, 1 , − ˆ ˆ = Ŵ(q α 1) +ˆ ˆ ˆ ∈ [− ] − + (27) α Ŵ(q 1) α α,α xD1 Lq(x) + (1 x)− Jq− (x), x 1, 1 . ˆ ˆ = Ŵ(q α 1) −ˆ ˆ ˆ ∈ [− ] − + We introduce the following rescale functions: 2x (a b) a, b 1, 1 x −b a+ ∧:[1 ] → [− ]: �→ (b− a)t a b 1, 1 a, b t − + + ∧− : [− ]→[ ]: �→ 2 and we write (x) as x . Te basis functions selected for the spatial discretization are given by­ 38,39: ∧ ˆ

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Figure 2. Plots of model (36) for β = 0.99 and α = 1.6.

2n 3 2 1,1 ϕn(x) Ln(x) Ln 2(x) + (1 x )Jn (x), x a, b . (28) = ˆ − + ˆ = 2(n 1) −ˆ ˆ ∈[ ] + 0 Te approximation space VN can be considered as follows: 0 V span ϕn(x), n 0, 1, ..., N 2 . (29) N = { = − } j 1 j 1 Te approximate solutions ψN+ and φN+ are shown as N 2 N 2 j 1 − j 1 j 1 − j 1 ψ + (x) ψ + ϕ (x), φ + (x) φ + ϕ (x), N = ˆ i i N = ˆi i (30) i 0 i 0 = = j 1 j 1 where ψ + and φ + are the unknown expansion coefcients to be determined. Choosing v ϕ ,0 k N 2 , ˆ i ˆi = k ≤ ≤ − the representation matrix of the Alikhanov L2-1σ spectral Legendre–Galerkin numerical scheme has the fol- lowing representation:

(β,σ,γ ) T j 1 j (β,σ,γ ) j 1 iM εσcαξj S S � + R1 σξj H1+ , + +  = − (β,σ,γ ) T  j 1 j (β,σ,γ ) j 1 (31) iM εσcαξ S S � R σξ H + , + j + + = 2 − j 2   where

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Figure 3. Plots of model (36) for β = 0.95 and α = 1.99.

j j j j T j j j j T � (ψ0, ψ1, ..., ψN 2) , � (φ0, φ1, ..., φN 2) , = ˆ ˆ ˆ − = ˆ ˆ ˆ − α α 2 2 N 2 sij aDx ϕi(x)xD ϕj(x)dx, S sij − , = b = i,j 0 �   = N 2 mij ϕi(x)ϕj(x)dx, M mij − , = = i,j 0 �   = j j 2 j 2 j j h1,i ϕi(x)IN k1 ψN (k1 2k2) φN ψN ξφN dx, = �    + +    +      j  j 2  j 2 j j h2,i ϕi(x)IN k1 φN (k1 2k2) ψN φN ξψN dx, (32) = �    + +    +      j j j j  T j  j j j T H1 (h1,0, h1,1, ..., h1,N 2) , H2 (h2,0, h2,1, ..., h2,N 2) , = − = − j j j (j,β,σ) j (j,β,σ) K d M�i, K d M�i, 1 = ˜i 2 = ˜i i 0 i 0 = = j (β,σ,γ ) T j (β,σ,γ ) j j R1 ε cα (1 σ)ξj S S � (1 σ)ξj H1 K1, =− −  +  − − − j (β,σ,γ ) T j (β,σ,γ ) j j R2 cαε(1 σ)ξj S S � (1 σ)ξj,2 H2 K2. =− −  +  − − −

Lemma 2.4 [see36,38] Te elements of the stifness matrix S in (31) are given by j j 2 j j 2 sij ai ai+ ai 2 ai+2, (33) = − − + + +

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Figure 4. Plots of model (36) for β = 0.6 and α = 1.99.

where

α α j 2 2 ai aDx Li(x)xDb Lj(x)dx = � ˆ ˆ

1 α N α α α α α α α α α α (34) b a − Ŵ(i 1)Ŵ(j 1) 2 , 2 2 , 2 2 , 2 2 , 2 2 , 2 − + + ̟r− − J − xr− − J− xr− − , =  2  Ŵ(i α 1)Ŵ(j α 1) i j 2 2 r 0     − + − + = α α α α N 2 , 2 2 , 2 and xr− − , ̟r− − are the Gauss-Jacobi points and their weights corresponding to the weight function α , α r 0 ω− 2 − 2 . Te mass matrix M= is symmetric and its nonzero elements are given as

b a b a 2j− 1 2j− 5 , i j,  + + + = mij mji (35) = =  b a 2j− 5 , i j 2. − + = + j 1,r j 1 j 1,r j 1,r  Also, as H + H + (ψ + , φ + ), p 1, 2 , r 0, the linear system (31) can be solved by Algorithm 1. p = p N N = ≥

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Figure 5. Plots of model (36) for β = 0.6 and α = 1.6.

Results and discussions In this section, we conduct numerical simulations to investigate the modeling capability of the space-time frac- tional Schrödinger equation (36). Computationally, we consider N 150 and M 1500. = = Example 1 We consider the following weakly coupled system:

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Figure 6. Plots of model (36) for β = 0.3 and α = 1.3.

β i ∂ ψ ( �)α/2ψ ψ 2 φ 2 ψ 0, ∂tβ − − + | | + | | =  � � (36)  β  i ∂ φ ( �)α/2φ φ 2 ψ 2 φ 0, ∂tβ − − + | | + | | =  � � with the initial conditions

iH0x ψ(x,0) p1sech p1x D e , = +  iH0x (37) φ(x,0) p2sech p2x D e− , = −  where p1 p2 1 , D 5 and H0 3. = = = = In the integer order case, i.e. for α 2 and β 1 , the wave propagates from the lef to the right spatial = = domain with a fxed angle. Te model is the Manakov system. Te collision is elastic and the system is completely integrable, the two waves cross each other, and their velocity and shape are unchanged. It is well known that the standard integer-order Schrödinger equation generates very difusive numerical solutions. Tis can be observed here when α is close to 2 and β is close to 1. Te numerical results in Figs. 1, 2, 3, 4 and 5 display the efect of the Lévy index 1 <α 2 and the temporal ≤ fractality 0 <β 1 on the shapes and stability of the soliton solutions. Tese results represent the performance ≤ of the space-time fractional Schrödinger equation (35). We fnd that when α 2 and β 1 , the collision is not = = elastic. Te non-integer orders signifcantly afect the shape of the solitons. One can see in Figs. 1 and 2 that, as the value of the space fractional-order α is decreased, the shape of the solitons changes more quickly. From Figs. 3, 4, 5 and 6, we note that the distinct selections of the fractional-order parameters β yield simu- lation results with diferent decay time or decay properties in the time direction. We observe that as the value of the time fractional-order β is decreased, the interface grows sharper and sharper at the beginning of time where-afer the solution propagates straight. Here, we notice a new phenomenon that as β is decreased, the decay

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Figure 7. Numerical simulations for Example 2 with γ = 0.05 and diferent values of the space and time fractional orders.

behavior is gradually reduced too. Te situation tends to the conventional case when β approaches 1. Tese fea- tures can be employed in physics to tunable sharpness of the space-time fractional Schrödinger equation (36) with the diferent choices of the space fractional order α and the time fractional order β , without changing the nonlinearity and dispersion efects.

Example 2 We consider the following strongly coupled system:

β i ∂ ψ ( �)α/2ψ ψ 2 φ 2 ψ γφ 0, x ( 25, 25), t (0, 40 , ∂tβ − − + | | + | | + = ∈ − ∈ ]  � � (38)  β  i ∂ φ ( �)α/2φ φ 2 ψ 2 φ γψ 0, x ( 25, 25), t (0, 40 , ∂tβ − − + | | + | | + = ∈ − ∈ ]  � � subject to the initial conditions

D H0 ψ(x,0) √2sech x ei 4 x, = + 2  (39) D i H0 x φ(x,0) √2sech x e− 4 , = − 2 

where D 25 and H0 1. = = It is well known that the linear coupling parameter γ dramatically afects the collision of solitary waves­ 40,41. In Figs. 7 and 8 we study the efect of the linear coupling parameter γ , the spatial fractional difraction order 1 <α 2 and the temporal fractality 0 <β 1 on the collision of solitary waves. We fnd when α tends to 2 and ≤ ≤ β tends to 1 that as the linear coupling parameter γ is increased, the jump behavior gets stronger. Moreover, for

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Figure 8. Numerical simulations for Example 2 with γ = 0.2 and diferent values of the space and time fractional orders.

any 1 <α 2 and when β tends to 1, the collision is always elastic. Te collision will occur earlier with increas- ≤ ing α and become closer to the classical integer-order case. For the space fractional Schrödinger equation with β 1 , the two soliton waves collide and keep their shapes when moving away afer the collision. Tis can be = observed here when β tends to 1. For smaller values of β , this feature is absent, the waves do not even cross, and the evolution of the solution is pretty sharp during the frst time steps. We note also that the distinct selections of the fractional-order parameters β yield simulation results with decay properties in the time direction, see Figs. 7 and 8 (second row). At the same time, some waves are created with oscillations corresponding to fuctuations and indicating the presence of decoherence phenomena that depends on β. Conclusion In this paper, we numerically investigated the nontrivial behavior of wave packets in a new type of time-space fractional quantum coupler. Te model involving a coupled system of time and space fractional nonlinear Schrödinger equations with linear or nonlinear coupling, can be used to uncover a wealth of information about an extended variety of phenomena, such as modeling the Bose-Einstein condensates, the interaction between pulses in nonlinear optics, or signals in nonlinear acoustic media. Te time-fractional derivative was considered to describe the nonlocal memory or decay behavior of the model. Te numerical simulation for modelling such kind of quantum couplers was carried using an easy to implement algorithm based on a novel consistent scheme. It is a combination of Galerkin spectral method of Legendre type, to approximate the spatial operators, and a high order fnite diference method, to approximate the temporal derivatives. Te efect of fractality parameters α and β on the behaviour of the solution is discussed. Te numerical simulations of the time-space fractional nonlinear Schrödinger equations in this paper show that the problem displays time decay behavior that has not been observed before in the time-fractional partial diferential equations modeling. Hence, the time and space fractional order operators can be used to control the decay behavior or the memory and the sharpness of the interface and undergo a seamless transition of the fractional order parameters. Of particular interest for this work also is the linear coupling parameter ξ . We test the efect of this parameter on the collision of solitary waves.

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Tese features can be employed in physics to tunable sharpness of the space-time fractional Schrödinger equa- tion with the diferent choices of the space fractional order α and the time fractional order β , without changing the nonlinearity and dispersion efects.

Received: 19 January 2021; Accepted: 27 April 2021

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Acknowledgements ASH wishes to acknowledge the support of RFBR Grant 19-01-00019. MAZ wishes to acknowledge the fnancial support of the National Research Centre of Egypt (NRC).

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Author contributions A.S.H. and M.A.Z. designed the project and wrote and edited the manuscript, and assisted with proofreading the manuscript. M.A.Z. and R.M.H. designed and carried out the experiments. A.S.H., M.A.Z. and R.H.D. ana- lysed the data, prepared fgures and wrote the original manuscript. R.H.D. provided expert advice, contributed to obtaining the funding and study design and proofread the manuscript. All coauthors contributed to editing the manuscript. Funding Tis study was supported fnancially by RFBR Grant (19-01-00019), the National Research Centre of Egypt (NRC) and Ghent university.

Competing interests Te authors declare no competing interests. Additional information Correspondence and requests for materials should be addressed to M.A.Z. Reprints and permissions information is available at www.nature.com/reprints. Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional afliations. Open Access Tis article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. Te images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://​creat​iveco​mmons.​org/​licen​ses/​by/4.​0/.

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