Structured light entities, chaos and nonlocal maps. A.Yu.Okulov∗ Russian Academy of Sciences, 119991, Moscow, Russian Federation. (Dated: July 26, 2019) Spatial chaos as a phenomenon of ultimate complexity requires the efficient numerical algorithms. For this purpose iterative low-dimensional maps have demonstrated high efficiency. Natural gener- alization of Feigenbaum and Ikeda maps may include convolution integrals with kernel in a form of Green function of a relevant linear physical system. It is shown that such iterative nonlocal nonlinear maps are equivalent to ubiquitous class of nonlinear partial differential equations of Ginzburg-Landau type. With a Green functions relevant to generic optical resonators these nonlocal maps emulate the basic spatiotemporal phenomena as spatial solitons, vortex eigenmodes breath- ing via relaxation oscillations mediated by noise, vortex-vortex and vortex-antivortex lattices with periodic location of vortex cores. The smooth multimode noise addition facilitates the selection of stable entities and elimination of numerical artifacts. PACS numbers: 05.45.-a 02.30.Rz 05.45.Ac 05.45.XT 05.45.Yv 05.65+b 42.30.Ms 42.60.Mi 42.65.Sf 42.81.Dp 47.27.De 47.32.C- Keywords: Discrete maps, integral transforms, soli- noise-mediated origin of complex patterns. Superconduc- tons, vortices, switching fronts, periodic solutions, vortex tors of type II in external magnetic field demonstrate the lattices, chaos, turbulence, probability density. hexagonal supercurrent lattices with nodal lines of pen- etrating magnetic induction B~ [5]. Thin slice nonlinear optical materials with feedback mirrors generate trian- I. INTRODUCTION gular intensity patterns [6]. Wide aperture solid state lasers emit coherent multivortex phase-locked beams of Complete spatial synchronization in nonlinear systems rectangular symmetry [7–9]. Delayed feedback systems is replaced by turbulent states with a fast decay of corre- of coupled oscillators perform reservoir computations [10] lations. The propagation dynamics ranges from switch- with Boolean ring networks [11]. ing waves to long-living localized excitations and spa- The transitions from regular patterns to turbulent tiotemporal solitons [1]. Numerical modeling of these states and vice versa are controlled by a limited num- complex nonlinear distributed systems is based upon ber of control parameters namely ambient temperature finite-difference schemes which emulate the basic solu- T, magnetic induction, density of carriers n, gain G and tions of partial differential equations such as localised loss γ, quadratic-qubic-quintic nonlinearities χ2,χ3,χ5, solitonic entities [2], propagation fronts, stable phase- carrier frequency ω = ck = 2π/λ, wavelength λ, dielec- locked periodic configurations alike lattices and self- tric permittivity ǫ , magnetic permeability µ and their organized vortex clusters. The alternative numerical ap- spatial distributions. Geometrical control parameters are proach is outlined. It is shown that discrete-time and system dimensions at different scales, aspect ratios (de- continuous space dynamical systems composed of se- gree of system asymmetry), say ratio of transverse di- quence of local nonlinear point-to-point Feigenbaum and mensions d, D to longitudinal extension L, focal lengths Ikeda-like maps [3] and nonlocal diffusion-dispersion in- of optical components F , frame of reference parameters tegral transforms are equivalent in a quite general set namely vectors of displacement velocity V~ and reference of cases to conventional partial differential equations of frame rotation angular velocity Ω~⊕ [12]. The variations Ginzburg-Landau type. Such nonlocal maps are itself of control parameters force system to switch between or- the robust numerical schemes that exhibit the transitions dered structures of different symmetry or toggle system from purely chaotic states to localization in momentum arXiv:1901.09274v4 [physics.optics] 25 Jul 2019 into spatially turbulent states with fast decay of correla- space inherent to corresponding coherent entities alike tions. The regions of stability are defined via calculation spatial solitons and vortex lattices in the presence of spa- of instability increments (Lyapunov exponents) [13]. tial noise. In some experimentally achievable cases the The above mentioned self-organized structures are pro- statistics generated by iterative maps is shown to have duced in framework of master equations of Ginzburg- the quantitative similarity with statistics of multimode Landau type. These evolution equations contain first or- random fields [4]. der derivative of the order parameter E over time t and 2 2 2 This computational approach is relevant to spatially ∂ ∂ ∂ diffusion terms ∆⊥ = 2 + 2 + 2 with second or- distributed nonlinear systems which demonstrate the ∂x ∂y ∂z der derivatives over spatial coordinates, so the simplest explicit finite difference numerical scheme has the form of iterative mapping, which mix the field values in adja- ∗ Electronic address: [email protected]; cent points of numerical mesh zm−1,zm,zm+1 separated URL: https://sites.google.com/site/okulovalexey by interval ∆z at each time step ∆t as: 2 The similar dynamics is embedded in Gross-Pitaevskii equation for macroscopic wavefunction Ψ of Bose- E +1 E n ,m − n,m = f(E )E + Einstein condensates [24]: ∆t n,m n,m En,m+1 + En,m−1 2En,m χ − . (1) 2 2 2 ∂Ψ(z, ~r, t) ~ 4π~ aS 2 ∆z i~ = ∆⊥Ψ+ U(~r, t)Ψ + Ψ Ψ. ∂t −2m m | | Behavior of such discrete dynamical system (DDS) (5) ranges from exactly solvable diffusion for purely real χ where U(~r, t) is confining potential of arbitrary com- and diffraction for purely imaginary χ towards spatiotem- plexity [25], aS is scattering length, m is mass of boson. poral instabilities [14], localized structures[15] and tur- In a presence of gain G and losses γ the NLS-GLE may bulence. The role of nonlinear term f(En,m) in pattern have a form of NLS with Frantz-Nodvik resonant gain formation is crucial [1, 16]. In many cases the numeri- term [26], relevant to amplification with stimulated emis- cal schemes (1) are replaced for much more sophisticated sion cross-section σ of light pulse of duration T2 <τ<T1 DDS of implicit type [17] or else convolution - like nonlo- in a rare earth doped dielectric with transverse T2 and cal integral transforms alike fast Fourier transform (FFT) longitudinal relaxation times T1: under a proper choice of spatial filtering [18–20]: ∂E(z, ~r, t) n ∂E i 2 ∞ + + ∆⊥E = ikn2 E E+ , , 3 , ∂z c ∂t 2k En+1(~r)= K(~r ~r )f(En(~r ))d ~r | | − ≈ t ′ Z−∞ 2 σNo(z, ~r)E exp[ 2σ E dt ] γE, (6) K(~r ~r )f(E (~r ))S(mx,my,mz), (2) − −∞ | | − − m n m Z mx,my,mz X We will discuss discrete iterative maps (2) for numeri- where S(mx,my,mz) is a finite volume element in- cal modeling of chaotic and regular spatiotemporal prop- stead of infinitesimal one d3~r, kernel K(~r ~r,) is a Green agation inherent to equations (3, 4, 5, 6). The article function of linear version of (1, 2), i.e. response− for delta- is organized as follows: section II is devoted to overview function δ(~r ~r,) in right part of scalar diffusion equation, of regular and chaotic iterative dynamics of real 1D [27] reads as − and complex point maps [28], section III outlines the di- K(~r ~r,) (χ/∆t)−1/2exp( √χ ~r ~r, 2/∆t) rect link between evolution PDEs (3-4) and maps (1-2), or for− scalar≈ parabolic diffraction− | equation− | this reads as section IV contains examples of localized solutions ob- K(~r ~r,) (χ/∆t)−1/2exp( i√χ ~r ~r, 2/∆t) [21]. tained with nonlocal maps being equivalent to equations Such− discrete≈ numerical approach− proved| − | to be extremely (3-4) , the spatially periodic and chaotic lattices are ob- effective for modeling of nonlinear dynamics inherent tained numerically in section V and nonlinear dynamics to evolution partial differential equations of so-called in the presence of multimode fluctuations is presented in parabolic type [22]. Apart from nonlocal dispersive section VI with conclusive discussion in section VII. laplacian there exists a nonlinear term f(En) which de- pends on square modulus of field E as in Kolmogorov- Petrovskii-Piskounov (KPP) equation [21]: II. ITERATIVE MAPS WITH UNIVERSAL BEHAVIOR. ∂E(z, ~r, t) 1 ∂E ∂2E Feigenbaum demonstrated the universality in itera- + + χ 2 = f(E)E. (3) ∂z V ∂t ∂t tions of real mapping with parabolic maximum of the Here the diffusion term χ = F 2/k2D2 may be respon- unit interval into itself alike logistic map [29]: sible for spatial filtering of high transverse harmonics in laser cavity by intracavity diaphragm and iteratively E +1 = λ E (1 E ), (7) repeated nonlinear transformation of the scalar field E n F n − n in ring laser with intracavity second harmonic genera- where the sole control parameter λF completely de- tion or Raman scattering [23]. More realistic models are termines the discrete time evolution of this simplest dy- based upon nonlinear Shrodinger equation, known also namical system. In particular he shown that two univer- as Ginzburg-Landau equation (NLS - GLE) which cap- sal irrational numbers, namely δF = 4.6692.. scales the tures the interplay of phase-amplitude modulation during separation of the values of λF = λ1, λ2, λ3...λM−1.λM ... propagation of complex field which is the source of mod- where period-doubling bifurcations occur(fig.1): ulational instability [22], solitons [2] and collapse [1]. For propagation of light pulse in Kerr dielectric NLS - GLE λM λM−1 reads as: δF = lim − 4.6692, (8) M→∞ λ −1 λ −2 → M − M ∂E(z, ~r, t) n ∂E i 2 and αF =2.5... scales the location of limit cycle points + + ∆⊥E = ikn2 E E. (4) ∂z c ∂t 2k | | in phase-space. 3 FIG. 2: a) Layout of unidirectional single transverse mode ring laser with nonlinear losses .