Domain walls in fractional media Shatrughna Kumar1 Pengfei Li23 and Boris A. Malomed14 1Department of Physical Electronics School of Electrical Engineering Faculty of Engineering

2025-05-03 0 0 2.16MB 13 页 10玖币

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Domain walls in fractional media

Shatrughna Kumar1, Pengfei Li2,3, and Boris A. Malomed1,4

1Department of Physical Electronics, School of Electrical Engineering, Faculty of Engineering,

and Center for Light-Matter Interaction, Tel Aviv University, P.O.B. 39040, Tel Aviv, Israel

2Department of Physics, Taiyuan Normal University, Jinzhong, 030619, China

32Institute of Computational and Applied Physics, Taiyuan Normal University, Jinzhong, 030619, China

4Instituto de Alta Investigaci´

on, Universidad de Tarapac´

a, Casilla 7D, Arica, Chile

Currently, much interest is drawn to the analysis of optical and matter-wave modes supported by the fractional

diffraction in nonlinear media. We predict a new type of such states, in the form of domain walls (DWs) in the

two-component system of immiscible ﬁelds. Numerical study of the underlying system of fractional nonlinear

Schr¨

odinger equations demonstrates the existence and stability of DWs at all values of the respective L´

evy

index (α < 2) which determines the fractional diffraction, and at all values of the XPM/SPM ratio βin the

two-component system above the immiscibility threshold. The same conclusion is obtained for DWs in the

system which includes the linear coupling, alongside the XPM interaction between the immiscible components.

Analytical results are obtained for the scaling of the DW’s width. The DW solutions are essentially simpliﬁed

in the special case of β= 3, as well as close to the immiscibility threshold. In addition to symmetric DWs,

asymmetric ones are constructed too, in the system with unequal diffraction coefﬁcients and/or different L´

evy

indices of the two components.

I. INTRODUCTION

The Schr¨

odinger equation with fractional spatial dispersion was originally derived for the wave function of particles moving by

L´

evy ﬂights, using the Feynman-integral formulation of fundamental quantum mechanics [1,2]. While experimental realization

of fractional quantum mechanics has not been reported yet, it was proposed to emulate it in terms of classical photonics, using

the commonly known similarity of the Schr¨

odinger equations and equations for the paraxial diffraction of optical beams [3,4].

A universal method for the emulation of the fractional diffraction is to use the basic 4fconﬁguration, which makes it possible

to perform the spatial Fourier transform of the beam, apply the phase shift, which is tantamount to the action of the fractional

diffraction, by means of an appropriate phase plate, and ﬁnally transform the beam back from the Fourier space [3]. In addition

to that, implementations of the fractional Schr¨

odinger equations were proposed in Levy crystals [5] and polariton condensates

[6].

Theoretical studies initiated by the above-mentioned scheme were developed in various directions, including the interplay

of the fractional diffraction with parity-time (PT ) symmetric potentials [7]-[10], propagation of Airy waves in the fractional

geometry [11,12], and adding the natural Kerr nonlinearity to the underlying setting, thus introducing fractional nonlinear

Schr¨

odinger equations (FNLSEs). The work with nonlinear models has produced many predictions, such as the modulational

instability of continuous waves (CWs) [13] and diverse types of optical solitons [14]-[44]. These are quasi-linear “accessible

solitons” [17,18], gap solitons maintained by lattice potentials [23]-[27], self-trapped vortices [28,29], multi-peak [30]-[33]

and cluster [34] modes, fractional solitons in discrete systems [35], localized states featuring spontaneously broken symmetry

[8,37,38], solitons in dual-core couplers [9,40], solitary states supported by the quadratic nonlinearity [41,42], and dark modes

[36]. Also studied were dissipative solitons in the fractional version of the complex Ginzburg-Landau equation [43]. Many of

these results were reviewed in Ref. [44].

The objective of the present work is to introduce one-dimensional settings for binary immiscible ﬁelds under the action of

the fractional diffraction. The immiscibility naturally gives rise to stable patterns in the form of domain walls (DW), alias grain

boundaries, which separate half-inﬁnite domains ﬁlled by the immiscible ﬁeld components. In areas of traditional physical

phenomenology, DWs are well known as basic patterns in thermal convection [45]-[49]. Grain boundaries of a different physical

origin occur in various condensed-matter settings [50]-[55]. In optics, DWs were predicted and experimentally observed in

bimodal light propagation in ﬁbers [56,57]. Similar states were predicted in binary Bose-Einstein condensates (BECs), provided

that the inter-component repulsion is stronger than the self-repulsion of each component, which provides for the immiscibility

[58]-[61].

The interplay of the two-component immiscibility, that maintains DWs, with fractional diffraction may naturally appear in

optics, considering the fractional bimodal propagation of light in a self-defocusing spatial waveguide. A similar model, based

on a system of fractional Gross-Pitaevskii equations (FGPEs) [44], may also naturally emerge in a binary BEC composed of

repulsively interaction particles which move by L´

evy ﬂights. We construct DW solutions for coupled FNLSEs and verify their

stability by means of numerical methods. Some results – in particular, scaling relations which determine the DW’s width as a

function of basic parameters of the system – are obtained in an analytical form.

The paper is organized as follows. The model is formulated in Section 2, which also includes analytical expressions for CW,

i.e., spatially uniform states, that may be linked by DW patterns, thus supporting them. Analytical results for the DWs are

arXiv:2210.09395v1 [physics.optics] 17 Oct 2022

collected in Section 3. Numerical results are reported in Section 4, and the paper is concluded by Section 5.

II. THE MODEL AND CW STATES

A. Basic equations

In terms of the optical bimodal propagation in the spatial domain, the scaled system of coupled FNLSEs for amplitudes of

copropagating electromagnetic waves u(x, z)and v(x, z)with orthogonal polarizations is

i∂u

∂z =1

2−∂2

∂x2α/2

u+ (|u|2+β|v|2)u−λv,

i∂v

∂z =1

2−∂2

∂x2α/2

v+ (|v|2+β|u|2)v−λu, (1)

where zis the propagation distance, xis the transverse coordinate, and the cubic terms, with normalized coefﬁcients 1and

β > 0, represent, respectively, the defocusing nonlinearity of the self-phase-modulation (SPM) and cross-phase-modulation

(XPM) types. The optical self-defocusing occurs, in particular, in semiconductor waveguides [62]. In the BEC model, the SPM

and XPM terms represent repulsive interactions between two atomic states in the binary condensate. In the latter case, the system

of scaled FGPEs is written in the form of Eq. (1), with zreplaced by the temporal variable, t.

In optics, two natural values of the XPM coefﬁcient are β= 2 for components uand vrepresenting circular polarizations

of light, or β= 2/3in the case of linear polarizations [63]. The value of βmay be varied in broader limits (in particular, the

case of β= 3 plays an essential role below) in photonic crystals [64,65]. In binary BEC, the effective XPM coefﬁcient can be

readily adjusted by means of the Feshbach resonance [66,67].

In the case of orthogonal linear polarizations in optics [corresponding to β= 2/3in Eq. (1)], the nonlinear interaction between

the components includes, in addition to the XPM terms, also the four-wave mixing (FWM), represented by terms (1/3) v2u∗

and (1/3) u2v∗in FNLSEs (1) for uand v(where ∗stands for complex conjugate), although these terms are usually suppressed

by the phase-velocity-birefringence effect [63]. In any case, the FWM terms appearing in the optical system with orthogonal

linear polarizations are not relevant in the present context, as the condition of the immiscibility of the two components holds

only for β > 1[see Eq. (12) below], eliminating the case of β= 2/3. The optical system with orthogonal circular polarizations

corresponds, as said above, to β= 2, which admits the immiscibility, but the FWM terms do not appear in the latter case.

Normally, they do not appear either in the BEC model based on the system of coupled FGPEs, therefore FWM terms are not

considered here.

The linear-coupling terms with coefﬁcient λ≥0in Eq. (1) account for mixing between the optical modes, or between the two

atomic states in BEC. In the former case, the linear mixing between circular polarizations may be imposed by the birefringence

[63], and in the latter case the mutual conversion of atomic states in BEC may be driven by resonant radiofrequency radiation

[68].

The fractional-diffraction operator with a positive L´

evy index (LI) αis deﬁned as the Riesz derivative [69–71],

−∂2

∂x2α/2

u(x)≡

2πZ+∞

−∞ |p|αdp Z+∞

−∞

dξeip(x−ξ)u(ξ)≡1

πZ+∞

pαdp Z+∞

−∞

dξ cos (p(x−ξ))u(ξ),(2)

which is built as the juxtaposition of the direct and inverse Fourier transform, with the fractional diffraction acting at the inter-

mediate stage. While there are different deﬁnitions of fractional derivatives, this one naturally appears in quantum mechanics

[1,2] and optics [3]. Normally, the LI takes values 1< α ≤2, but, in the case of the self-defocusing sign of the nonlinearity,

when the system is not subject to the wave collapse (implosion driven by self-attraction), it is also possible to consider values

0< α ≤1. The usual (non-fractional) diffraction naturally corresponds to α= 2 in Eq. (1).

Stationary solutions to Eqs. (1) with propagation constant k < 0are looked for as

{u(x, z), v (x, z)}=eikz {U(x), V (x)},(3)

where U(x)and V(x)are real functions which satisfy the following system of equations:

kU +1

2−∂2

∂x2α/2

U+ (U2+βV 2)U−λV = 0,

kV +1

2−∂2

∂x2α/2

V+ (V2+βU2)V−λU = 0.(4)

The energy (Hamiltonian) of the stationary state (4) with the Riesz derivatives deﬁned as per Eq. (2) is

E=1

4πZ+∞

−∞

dx Z+∞

−∞

dξ Z+∞

pαdp Z+∞

−∞

dξ cos (p(x−ξ)) [U(x)U(ξ) + V(x)V(ξ)]

+Z+∞

−∞

dx 1

4U4+V4+ 2βU2V2−λUV .(5)

Stability of stationary solutions, obtained in the form of expression (3), against small perturbations was investigated by means

of the usual approach, looking for the perturbed solution as

u(x, z) = eikz hU(x) + eγz a(x) + eγ∗zb∗(x)i,

v(x, z) = eikz hV(x) + eγz c(x) + eγ∗zd∗(x)i,(6)

where {a(x), b(x), c(x), d(x)}are components of an eigenmode of inﬁnitesimal perturbations, and γis the respective eigenvalue

(which may be a complex number). The substitution of the perturbed expression (6) in Eq. (1) and linearization leads to the

system of coupled equations,

(−k+iγ)a=1

2−∂2

∂x2α/2

a+2U2+βV 2a+U2b+βUV (c+d)−λc,

(−k−iγ)b=1

2−∂2

∂x2α/2

b+2U2+βV 2b+U2a+βUV (c+d)−λd,

(7)

(−k+iγ)c=1

2−∂2

∂x2α/2

c+2U2+βV 2c+U2d+βUV (a+b)−λa,

(−k−iγ)d=1

2−∂2

∂x2α/2

d+2U2+βV 2d+U2c+βUV (a+b)−λb.

The underlying DW solution is stable if numerical solution of Eq. (7) yields solely imaginary eigenvalues, with zero real parts.

B. Continuous-wave (CW) solutions and the immiscibility condition

The spatially uniform version of Eq. (4), with U, V = const, gives rise to two asymmetric (partly immiscible, with U6=V)

CW solutions, labeled by subscripts +and −, which are mirror images of each other:

U+

V+=1

√2





q−k

2+λ

β−1+q−k

2−λ

β−1

q−k

2+λ

β−1−q−k

2−λ

β−1







,(8)

U−

V−=1

√2





q−k

2+λ

β−1−q−k

2−λ

β−1

q−k

2+λ

β−1+q−k

2−λ

β−1







.(9)

Note that the total density of solutions (8) and (9) is

++V2

+=U2

−+V2

−=−k. (10)

While in the limit of λ= 0 (no linear mixing), the obvious CW states are completely immiscible, with V+=U−= 0, the partly

immiscible states given by Eqs. (8) and (9) were found only recently in Ref. [49]. Parallel to the asymmetric CW states (8) and

(9) there is the mixed (symmetric) one, with

U0=V0=p(λ−k)/(1 + β).(11)

For given k, i.e., for given CW density [see Eq. (10)], CW states (8) and (9) exist under the following condition:

β−1>(β−1)immisc ≡2λ/|k|.(12)

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DomainwallsinfractionalmediaShatrughnaKumar1,PengfeiLi2;3,andBorisA.Malomed1;41DepartmentofPhysicalElectronics,SchoolofElectricalEngineering,FacultyofEngineering,andCenterforLight-MatterInteraction,TelAvivUniversity,P.O.B.39040,TelAviv,Israel2DepartmentofPhysics,TaiyuanNormalUniversity,Jinzhong,0306...

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Domain walls in fractional media Shatrughna Kumar1 Pengfei Li23 and Boris A. Malomed14 1Department of Physical Electronics School of Electrical Engineering Faculty of Engineering

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