The impact of memory effect on space fractional strong quantum couplers with tunable decay behavior and its numerical simulation

The nontrivial behavior of wave packets in the space fractional coupled nonlinear Schrödinger equation has received considerable theoretical attention. The difficulty comes from the fact that the Riesz fractional derivative is inherently a prehistorical operator. 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 finite-difference/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

A novel, structure-preserving, second-order-in-time relaxation scheme for Schrödinger-Poisson systems

The authors acknowledge the support from The Carnegie Trust Research Incentive Grant RIG008215 . I.K. would also like to acknowledge the support from London Mathematical Society through an Emmy Noether Fellowship . In addition, Th. K. and I.K. thank the Edinburgh Mathematical Society for the Covid Recovery Fund that allowed for the completion and the submission of this paper.We introduce a new structure preserving, second order in time relaxation-type scheme for approximating solutions of the Schrödinger-Poisson system. More specifically, we use the Crank-Nicolson scheme as a time stepping mechanism, whilst the nonlinearity is handled by means of a relaxation approach in the spirit of [10,11,34] for the nonlinear Schrödinger equation. For the spatial discretisation we use the standard conforming finite element scheme. The resulting scheme is explicit with respect to the nonlinearity, i.e. it requires the solution of a linear system for each time-step, and satisfies discrete versions of the system's mass conservation and energy balance laws for constant meshes. The scheme is seen to be second order in time. We conclude by presenting some numerical experiments, including an example from cosmology and an example with variable time-steps which demonstrate the effectiveness and robustness of the new scheme.Peer reviewe

B-Spline Collocation Methods For Coupled Nonlinear Schrödinger Equation

In this study, the Coupled Nonlinear Schrödinger Equation (CNLSE) which models the propagation of light waves in optical fiber is solved using numerical methods namely Finite Difference Method (FDM) and B-Spline collocation methods. The equation was discretized in space and time. We propose the discretization of the nonlinear terms in the CNLSE following the Taylor approach and a newly developed approach called Besse. The theta-weighted method is used to generalize the scheme whereby the Crank-Nicolson scheme (i.e θ = 0.5) is chosen. The time derivatives are discretized by forward difference approximation. For each approach, the space dimension is then discretized by five different collocation methods independently. The first method for Taylor approach is based on FDM whereby the space derivatives are replaced by central difference approximation

The impact of memory effect on space fractional strong quantum couplers with tunable decay behavior and its numerical simulation

The nontrivial behavior of wave packets in the space fractional coupled nonlinear Schrödinger equation has received considerable theoretical attention. The difficulty comes from the fact that the Riesz fractional derivative is inherently a prehistorical operator. 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 finite-difference/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. © 2021, The Author(s).This study was supported financially by RFBR Grant (19-01-00019), the National Research Centre of Egypt (NRC) and Ghent university

Petrov-Galerkin Method for the Coupled Schrödinger-KdV Equation

Petrov-Galerkin method is used to derive a numerical scheme for the coupled Schrödinger-KdV (SKdV) equations, where we have used the cubic B-splines as a test functions and a linear B-splines as a trial functions. Product approximation technique is used to deal with the nonlinear terms. An implicit midpoint rule and the Runge-Kutta method of fourth-order (RK4) are used to discretize in time. A block nonlinear pentadiagonal system is obtained. We solve this system by the fixed point method. The resulting scheme has a fourth-order accuracy in space direction and second-order in time direction in case of the implicit midpoint rule and it is unconditionally stable by von Neumann method. Using the RK4 method the scheme will be linear and fourth-order in time and space directions, and it is also conditionally stable. The exact soliton solution and the conserved quantities are used to assess the accuracy and to show the robustness and the efficiency of the proposed schemes

Unconditionally stable integration of Maxwell's equations

Numerical integration of Maxwell’s equations is often based on explicit methods accepting a stability step size restriction. In literature evidence is given that there is also a need for unconditionally stable methods, as exemplified by the successful alternating direction implicit – finite difference time domain scheme. In this paper, we discuss unconditionally stable integration for a general semidiscrete Maxwell system allowing non-Cartesian space grids as encountered in finite-element discretizations. Such grids exclude the alternating direction implicit approach. Particular attention is given to the second-order trapezoidal rule implemented with preconditioned conjugate gradient iteration and to second-order exponential integration using Krylov subspace iteration for evaluating the arising ϕ-functions. A three-space dimensional test problem is used for numerical assessment and comparison with an economical second-order implicit–explicit integrator

Large deviations for rare realizations of dynamical systems

A central problem in uncertainty quantification is how to characterize the impact that our incomplete knowledge about models has on the predictions we make from them. This question naturally lends itself to a probabilistic formulation, by making the unknown model parameters random with given statistics. In the following this approach is used in concert with tools from large deviation theory (LDT) and optimal control to estimate the probability that some observables in a dynamical system go above a large threshold after some time, given the prior statistical information about the system’s parameters and/or its initial conditions. It is established under which conditions the extreme events occur in a predictable way, as the minimizer of the LDT action functional, i.e. the instanton. In the first physical application, the appearance of rogue waves in a long-crested deep sea is investigated. First, the leading order equations are derived for the wave statistics in the framework of wave turbulence (WT), showing that the theory cannot go beyond Gaussianity, although it remains the main tool to understand the energetic transfers. It is shown how by applying our LDT method one can use the incomplete information contained in the spectrum (with the Gaussian statistics of WT) as prior and supplement this information with the governing nonlinear dynamics to reliably estimate the probability distribution of the sea surface elevation far in the tail at later times. Our results indicate that rogue waves occur when the system hits unlikely pockets of wave configurations that trigger large disturbances of the surface height. The rogue wave precursors in these pockets are wave patterns of regular height but with a very specific shape that is identified explicitly, thereby potentially allowing for early detection. Finally, the first experimental evidence of hydrodynamic instantons is presented using data collected in a long wave flume, elevating the instanton description to the role of a unifying theory of extreme water waves. Other applications of the method are illustrated: To the nonlinear Schrödinger equation with random initial conditions, relevant to fiber optics and integrable turbulence, and to a rod with random elasticity pulled by a time-dependent force. The latter represents an interesting nonequilibrium statistical mechanics setup with a strongly out-of-equilibrium transient (absence of local thermodynamic equilibrium) and a small number of degrees of freedom (small system), showing how the LDT method can be exploited to solve optimal-protocol problems

Nonlinear edge waves in mechanical topological insulators

We show theoretically that the classical 1D nonlinear Schrödinger (NLS) and coupled nonlinear Schrödinger (CNLS) equations govern the envelope(s) of localised and unidirectional nonlinear travelling edge waves in a 2D mechanical topological insulator (MTI). The MTI consists of a collection of pendula with weak Duffing nonlinearity connected by linear springs that forms a mechanical analogue of the quantum spin Hall effect (QSHE). It is found, through asymptotic analysis and dimension reduction, that the NLS and CNLS respectively describe the unimodal and bimodal properties of the nonlinear system. The governing bimodal CNLS is found to be non-integrable by nature and as such we discover new solutions by exploring the spatial dynamics of the reduced travelling wave ODE with general parameters. Such solutions include travelling fronts and, by numerically continuing these fronts, one can find vector soliton (VS) in non integrable CNLS equations. The equilibria can also undergo both pitchfork and Turing bifurcation in the reversible spatial dynamical system and we discuss relevant conditions for the existence and consequences of such critical values. We briefly discuss the necessity of the developed front condition in forming such structures and present an analytical framework for front-grey soliton collisions by utilising conserved quantities of the non-integrable CNLS. The existence/stability of front and VS solutions can be inferred by spatial hyperbolicity and linear stability of the background fields, with the criteria presented here. VS solutions are considered in the form of bright-bright, bright-dark, and dark-dark solitons and their collision dynamics are explored qualitatively in the non-integrable regime. The Turing analysis presents the existence of periodic and localised patterned states in the CNLS, and we compare these solutions to those found in the analysis of the Swift-Hohenberg equation. Theoretical predictions from the 1D (C)NLS are confirmed by numerical simulations of the original 2D MTI for various types of travelling waves and rogue waves. As a result of topological protection the edge solitons persist over long time intervals and through irregular boundaries. Due to the robustness of topologically protected edge solitons (TPES) it is suggested that their existence may have significant implications on the design of acoustic devices. Spacetime simulations show a clear possibility of utilising MTIs in acoustical cloaking with TPES a vital player in such processes

Wave dynamics in random, absorptive or laseractive media

We consider the behavior of light propagating in dielectrically disordered and energetically nonconservative material. Disorder and energy nonconservation can be dealt with via the use of the mathematical formalism commonly known as the Keldysh technique. We derive in the Keldysh formalism a field theory of light propagation in disordered, nonconservative media. This field theoretical formulation is commonly known as the nonlinear sigma model. We also show how to calculate physical quantities like correlation functions from the sigma model, and how a source term can be included in the action of the field theory. We apply the derived field theory to the calculation of full counting statistics. We derive a generating functional for the cumulants of energy transmitted through a weakly nonconservative one-dimensional disordered system. We find fluctuations of transmittance which is in accordance to Dorokhov’s distribution of transmission coefficients. Our numerical results also agree quantitatively with previous diagrammatic results of low order cumulants. We apply the field theoretical formalism to random lasing. We calculate the photonic distribution function. We find that the distribution function obeys a nonlocal Fisher equation. Finally we consider the effect of the vector nature of light on wave properties, specifically whether polarization increases or decreases the propensity of light waves in disordered dielectric media to become localized (Anderson localization).We map the light polarization to a “pseudospin” degree of freedom which we then treat with techniques adapted from classical studies of electronic spin. We find that the polarization of light waves does in fact contribution to a diminished probability of return to the origin, the value of which determines of course the ease for the occurrence of Anderson localization