Numerical solution to coupled nonlinear Schrödinger equations on unbounded domains

The numerical simulation of coupled nonlinear Schrödinger equations on unbounded domains is considered in this paper. By using the operator splitting technique, the original problem is decomposed into linear and nonlinear subproblems in a small time step. The linear subproblem turns out to be two decoupled linear Schrödinger equations on unbounded domains, where artificial boundaries are introduced to truncate the unbounded physical domains into finite ones. Local absorbing boundary conditions are imposed on the artificial boundaries. On the other hand, the coupled nonlinear subproblem is an ODE system, which can be solved exactly. To demonstrate the effectiveness of our method, some comparisons in terms of accuracy and computational cost are made between the PML approach and our method in numerical examples.     

The focusing nonlinear Schrödinger equation: Effect of the coupling to a low frequency field

The main properties of the focusing nonlinear Schrödinger equation for a multidimensional wavepacket envelop are reviewed. The coupling to a mean field or to low frequency acoustic waves stirred by the wavepacket is analyzed.     

Explicit approximate controllability of the Schrödinger equation with a polarizability term

We consider a controlled Schrödinger equation with a dipolar and a polarizability term, used when the dipolar approximation is not valid. The control is the amplitude of the external electric field, it acts nonlinearly on the state. We extend in this infinite dimensional framework previous techniques used by Coron, Grigoriu, Lefter and Turinici for stabilization in finite dimension. We consider a highly oscillating control and prove the semi-global weak $H^2$ stabilization of the averaged system using a Lyapunov function introduced by Nersesyan. Then it is proved that the solutions of the Schrödinger equation and of the averaged equation stay close on every finite time horizon provided that the control is oscillating enough. Combining these two results, we get approximate controllability to the ground state for the polarizability system with explicit controls. Numerical simulations are presented to illustrate those theoretical results.     

Lattice Quantum Algorithm for the Schrödinger Wave Equation in 2+1 Dimensions with a Demonstration by Modeling Soliton Instabilities

A lattice-based quantum algorithm is presented to model the non-linear Schr?dinger-like equations in 2 + 1 dimensions. In this lattice-based model, using only 2 qubits per node, a sequence of unitary collide (qubit–qubit interaction) and stream (qubit translation) operators locally evolve a discrete field of probability amplitudes that in the long-wavelength limit accurately approximates a non-relativistic scalar wave function. The collision operator locally entangles pairs of qubits followed by a streaming operator that spreads the entanglement throughout the two dimensional lattice. The quantum algorithmic scheme employs a non-linear potential that is proportional to the moduli square of the wave function. The model is tested on the transverse modulation instability of a one dimensional soliton wave train, both in its linear and non-linear stages. In the integrable cases where analytical solutions are available, the numerical predictions are in excellent agreement with the theory.     

A note to the integrable discretization of the mKdV and Schrödinger equations

An approach has been proposed to the integrable discretization of nonlinear evolution equations. Based on the bilinear formalism, we choose appropriate substitution from hyperbolic operator into continuous Hirota operators and obtain several new kinds of integrable system through seeking their 3-soliton solutions, such as the mKdV equation, the nonlinear Schrödinger equation and so on. By applying Adomian decompose method, we discuss the numerical analysis property to the discrete mKdV equation. In addition, we also point out the relations between the above discreted equations and some well-known equations.     

Numerical simulation of coupled nonlinear Schrödinger equation

The coupled nonlinear Schrödinger equation models several interesting physical phenomena. It represents a model equation for optical fiber with linear birefringence. In this paper we introduce a finite difference method for a numerical simulation of this equation. This method is second-order in space and conserves the energy exactly. It is quite accurate and describes the interaction picture clearly according to our numerical results.     

Symplectic integrators for discrete nonlinear Schrödinger systems

Symplectic methods for integrating canonical and non-canonical Hamiltonian systems are examined. A general form for higher order symplectic schemes is developed for non-canonical Hamiltonian systems using generating functions and is directly applied to the Ablowitz–Ladik discrete nonlinear Schrödinger system. The implicit midpoint scheme, which is symplectic for canonical systems, is applied to a standard Hamiltonian discretization. The symplectic integrators are compared with an explicit Runge–Kutta scheme of the same order. The relative performance of the integrators as the dimension of the system is varied is discussed.     

The well-posedness for fractional nonlinear Schrödinger equations

In this paper we apply the tools of harmonic analysis to study the Cauchy problem for time fractional Schrödinger equations. The existence and a sharp decay estimate for solutions of the given problem in two different spaces are addressed. Some fundamental properties of operators appearing in the solution of the problem are also discussed.     

Finite difference method for time-space-fractional Schrödinger equation

In this paper, an implicit finite difference scheme for the nonlinear time-space-fractional Schrödinger equation is presented. It is shown that the implicit scheme is unconditionally stable with experimental convergence order of O(τ^2?α+h²), where τ and h are time and space stepsizes, respectively, and α (0<α<1) is the fractional-order in time. In order to reduce the computational cost, the explicit–implicit scheme is proposed such that the nonlinear term is easily treated. Meanwhile, the implicit finite difference scheme for the coupled time-space-fractional Schrödinger system is also presented, which is unconditionally stable too. Numerical examples are given to support the theoretical analysis.     

Multiple positive solutions for a Schrödinger–Newton system with sign-changing potential

In this paper, we study the Schrödinger–Newton systems with sign-changing potential in a bounded domain. By using the variational method and analytic techniques, the existence and multiplicity of positive solutions are established.     

New schemes for the coupled nonlinear Schrödinger equation

In this paper, we present three new schemes for the coupled nonlinear Schrödinger equation. The three new schemes are multi-symplectic schemes that preserve the intrinsic geometry property of the equation. The three new schemes are also semi-explicit in the sense that they need not solve linear algebraic equations every time-step, which is usually the most expensive in numerical simulation of partial differential equations. Many numerical experiments on collisions of solitons are presented to show the efficiency of the new multi-symplectic schemes.     

Numerical analysis of a new conservative scheme for the coupled nonlinear Schrödinger equations

We devote the present paper to an efficient conservative scheme for the coupled nonlinear Schrödinger (CNLS) system, based on the Fourier pseudospectral method, the Crank–Nicolson method and leap-frog method. To obtain the present scheme, the key idea consists of two aspects. First, we solve the CNLS system based on its Hamiltonian structure and the resulted scheme can preserve the Hamiltonian nature. Second, we use Fourier pseudospectral method in spatial discretization and Crank–Nicolson/ leap-frog scheme for discretizing linear/ nonlinear terms in time direction, respectively. The proposed scheme is energy-preserving, mass-preserving, uniquely solvable and unconditionally stable, while being decoupled, linearized and suitable for parallel computation in practical computation. Using the energy method and the classical interpolation theory, an error estimate of the proposed scheme is proven strictly without any grid ratio restrictions in the discrete L² norm. Finally, numerical results are reported to verify our theoretical analysis.     

Stabilisation of Schrödinger equation in dynamic boundary feedback with a memory-typed heat equation

In this work, we study the dynamic behaviour for a heat equation with exponential polynomial kernel memory to be a controller for a Schrödinger system. By introducing some new variables, the time-variant system is transformed into a time-invariant one. Remarkably, the resolvent of the closed-loop system operator is not compact anymore. The residual spectrum is shown to be empty and the continuous spectrum consisting of finite isolated points are obtained. It is shown that the sequence of generalised eigenfunctions forms a Riesz basis for the Hilbert state space. This deduces the spectrum-determined growth condition for the C ₀-semigroup, and the exponential stability is then established.     

Variable preconditioning in complex Hilbert space and its application to the nonlinear Schrödinger equation

An iterative method is developed for nonlinear equations in complex Hilbert spaces, extending the method of variable preconditioning defined earlier in real spaces. We derive convergence of our method. The motivating example for this extension is the time-dependent nonlinear Schrödinger equation, where we use our iteration for the time discretization of the problem and test it numerically.     

Nonlinear grid mapping applied to an FDTD-based,multi-center 3D Schrödinger equation solver

We developed a straightforward yet effective method of increasing the accuracy of grid-based partial differential equation (PDE) solvers by condensing computational grid points near centers of interest. We applied this “nonlinear mapping” of grid points to a finite-differenced explicit implementation of a time-dependent Schrödinger equation solver in three dimensions. A particular multi-center mapping was developed for systems with multiple Coulomb potentials, allowing the solver to be used in complex configurations where symmetry cannot be used for simplification. We verified our method by finding the eigenstates and eigenenergies of the hydrogen atom and the hydrogen molecular ion (${\mathrm{H}}_2^{+}$) and comparing them to known solutions. We demonstrated that our nonlinear mapping scheme – which can be readily added to existing PDE solvers – results in a marked increase in accuracy versus a linear mapping with the same number of (or even much fewer) grid points, thus reducing memory and computational requirements by orders of magnitude.     

On the L∞ convergence of a difference scheme for coupled nonlinear Schrödinger equations

In this article, a finite difference scheme for coupled nonlinear Schrödinger equations is studied. The existence of the difference solution is proved by Brouwer fixed point theorem. With the aid of the fact that the difference solution satisfies two conservation laws, the finite difference solution is proved to be bounded in the discrete $L_{\infty }$ norm. Then, the difference solution is shown to be unique and second order convergent in the discrete $L_{\infty }$ norm. Finally, a convergent iterative algorithm is presented.     

Algebraic techniques for Schrödinger equations in split quaternionic mechanics

The split quaternionic Schrödinger equation $\frac{?}{?t}|f〉=?A|f〉$ plays an important role in split quaternionic mechanics, in which $A$ a split quaternion matrix. This paper, by means of a real representation of split quaternion matrices, studies problems of split quaternionic Schrödinger equation, and gives an algebraic technique for the split quaternionic Schrödinger equation. This paper also derives an algebraic technique for finding eigenvalues and eigenvectors of a split quaternion matrix in split quaternionic mechanics.     

Numerical solution of two-dimensional Schrödinger equation by Boadway's transformation

In this article, Boadway's transformation technique is extended to the two-dimensional Schrödinger equation. The result of numerical experiments is presented, and a comparison of the present results with the exact solution shows an extremely good agreement.