Efficient approximation algorithm for the Schrödinger–Possion system

In this article, we study an efficient approximation algorithm for the Schrödinger–Possion system arising in the resonant tunneling diode (RTD) structure. By following the classical Gummel iterative procedure, we first decouple this nonlinear system and prove the convergence of the iteration method. Then via introducing a novel spatial discrete method, we solve efficiently the decoupled Schrödinger and Possion equations with discontinuous coefficients on no‐uniform meshes at each iterative step, respectively. Compared with the traditional ones, the algorithm considered here not only has a less restriction on the discrete mesh, but also is more accurate. Finally, some numerical experiments are shown to confirm the efficiency of the proposed algorithm.     

Numerical solutions of nonlinear fractional Schrödinger equations using nonstandard discretizations

In this article, numerical study for both nonlinear space‐fractional Schrödinger equation and the coupled nonlinear space‐fractional Schrödinger system is presented. We offer here the weighted average nonstandard finite difference method (WANSFDM) as a novel numerical technique to study such kinds of partial differential equations. The space fractional derivative is described in the sense of the quantum Riesz‐Feller definition. Stability analysis of the proposed method is studied. To show that this method is reliable and computationally efficient different numerical examples are provided. We expect that the proposed schemes can be applicable to different systems of fractional partial differential equations. © 2016 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 1399–1419, 2017     

Dunkl-Schrödinger semigroups and applications

We obtain dispersive estimates for the linear Dunkl–Schrödinger equations with and without quadratic potential. As a consequence, we prove the local well-posedness for semilinear Dunkl–Schrödinger equations with polynomial nonlinearity in certain magnetic field. Furthermore, we study many applications: as the uncertainty principles for the Dunkl transform via the Dunkl–Schrödinger semigroups, the embedding theorems for the Sobolev spaces associated with the generalized Hermite semigroup. Finally, almost every where convergence of the solutions of the Dunkl–Schrödinger equation is also considered.     

Homotopy perturbation method for coupled Schrödinger–KdV equation

In this paper, numerical analysis of the coupled Schrödinger–KdV equation is studied by using the Homotopy Perturbation Method (HPM). The available analytical solutions of the coupled Schrödinger–KdV equation obtained by multiple traveling wave method are compared with HPM to examine the accuracy of the method. The numerical results validate the convergence and accuracy of the Homotopy Perturbation Method for the analyzed coupled Schrödinger–KdV equation.     

Extended Lagrangian approach for the defocusing nonlinear Schrödinger equation

We study the defocusing nonlinear Schrödinger (NLS) equation written in hydrodynamic form through the Madelung transform. From the mathematical point of view, the hydrodynamic form can be seen as the Euler–Lagrange equations for a Lagrangian submitted to a differential constraint corresponding to the mass conservation law. The dispersive nature of the NLS equation poses some major numerical challenges. The idea is to introduce a two‐parameter family of extended Lagrangians, depending on a greater number of variables, whose Euler–Lagrange equations are hyperbolic and accurately approximate NLS equation in a certain limit. The corresponding hyperbolic equations are studied and solved numerically using Godunov‐type methods. Comparison of exact and asymptotic solutions to the one‐dimensional cubic NLS equation (“gray” solitons and dispersive shocks) and the corresponding numerical solutions to the extended system was performed. A very good accuracy of such a hyperbolic approximation was observed.     

Discontinuous Galerkin and multiscale variational schemes for a coupled damped nonlinear system of Schrödinger equations

In this article, we study a streamline diffusion‐based discontinuous Galerkin approximation for the numerical solution of a coupled nonlinear system of Schrödinger equations and extend the resulting method to a multiscale variational scheme. We prove stability estimates and derive optimal convergence rates due to the maximal available regularity of the exact solution. In the weak formulation, to make the underlying bilinear form coercive, it was necessary to supply the equation system with an artificial viscosity term with a small coefficient of order proportional to a power of mesh size. We justify the theory by implementing an example of an application of the time‐dependent Schrödinger equation in the coupled ultrafast laser. © 2013 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013     

Dispersive properties of a viscous numerical scheme for the Schrödinger equation

In this Note we study the dispersive properties of the numerical approximation schemes for the free Schrödinger equation. We consider finite-difference space semi-discretizations. We first show that the standard conservative scheme does not reproduce at the discrete level the properties of the continuous Schrödinger equation. This is due to spurious high frequency numerical solutions. In order to damp out these high-frequencies and to reflect the properties of the continuous problem we add a suitable extra numerical viscosity term at a convenient scale. We prove that the dispersive properties of this viscous scheme are uniform when the mesh-size tends to zero. Finally we prove the convergence of this viscous numerical scheme for a class of nonlinear Schrödinger equations with nonlinearities that may not be handeled by standard energy methods and that require the so-called Strichartz inequalities. To cite this article: L.I. Ignat, E. Zuazua, C. R. Acad. Sci. Paris, Ser. I 340 (2005).     

Conservative finite difference methods for fractional Schrödinger–Boussinesq equations and convergence analysis

In this paper, two conservative finite difference schemes for fractional Schrödinger–Boussinesq equations are formulated and investigated. The convergence of the nonlinear fully implicit scheme is established via discrete energy method, while the linear semi‐implicit scheme is analyzed by means of mathematical induction method. Our schemes are proved to preserve the total mass and energy in discrete level. The numerical results are given to confirm the theoretical analysis.     

A novel numerical approach to simulating nonlinear Schrödinger equations with varying coefficients

We describe a novel numerical approach to simulations of nonlinear Schrödinger equations with varying coefficients, based on the discovery of a new and intrinsic conservation law for varying coefficient nonlinear Schrödinger equations. The approach is shown to preserve some crucial classical conservations, such as the spatial ergodicity, and utilized in numerical simulations of periodically and quasi-periodically solitary waves for nonlinear Schrödinger equations with periodic or quasi-periodic coefficients. Some numerical experiments are presented to illustrate the conservative property.     

An application of exp‐function method to approximate general and explicit solutions for nonlinear Schrödinger equations

In this work, we implement a relatively new analytical technique, the exp‐function method, for solving nonlinear equations and absolutely a special form of generalized nonlinear Schrödinger equations, which may contain high‐nonlinear terms. This method can be used as an alternative to obtain analytical and approximate solutions of different types of fractional differential equations, which is applied in engineering mathematics. Some numerical examples are presented to illustrate the efficiency and the reliability of exp‐function method. It is predicted that exp‐function method can be found widely applicable in engineering. © 2010 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 27: 1016–1025, 2011     

A two-grid approximation scheme for nonlinear Schrödinger equations: dispersive properties and convergence

We introduce a two-grid finite difference approximation scheme for the free Schrödinger equation. This scheme is shown to converge and to posses appropriate dispersive properties as the mesh-size tends to zero. A careful analysis of the Fourier symbol shows that this occurs because the two-grid algorithm (consisting in projecting slowly oscillating data into a fine grid) acts, to some extent, as a filtering one. We show that this scheme converges also in a class of nonlinear Schrödinger equations whose well-posedness analysis requires the so-called Strichartz estimates. This method provides an alternative to the method introduced by the authors [L.I. Ignat, E. Zuazua, Dispersive properties of a viscous numerical scheme for the Schrödinger equation, C. R. Math. Acad. Sci. Paris 340 (7) (2005) 529–534] using numerical viscosity. To cite this article: L.I. Ignat, E. Zuazua, C. R. Acad. Sci. Paris, Ser. I 341 (2005).     

Réflexion des Singularités pour L'équation de Schrödinger

Abstract

RÉSUMÉ Nous établissons un lien entre la solution de l'équation de Schrödinger avec conditions de Dirichlet et une équation hyperbolique pour laquelle on peut appliquer les résultats classiques de réflexion des singularités, ce qui nous permet de prouver des résultats de réflexion des singularités pour l'équation de Schrödinger. Enfin, on utilise ces résultats pour calculer l'opérateur de Neumann associé à l'équation de Schrödinger.

A link between the Schrödinger equation and an hyperbolic equation is shown. Classical results on reflection of singularities for hyperbolic equations allow to derive reflection of singularities results for the Schrödinger equation. Finally, these results are used to compute the Neumann operator for the Schrödinger equation.     

Stochastic Nonlinear Schrödinger Equations with Linear Multiplicative Noise: Rescaling Approach

We prove well-posedness results for stochastic nonlinear Schrödinger equations with linear multiplicative Wiener noise, including the nonconservative case. Our approach is different from the standard literature on stochastic nonlinear Schrödinger equations. By a rescaling transformation we reduce the stochastic equation to a random nonlinear Schrödinger equation with lower-order terms and treat the resulting equation by a fixed point argument based on generalizations of Strichartz estimates proved by Marzuola et al. (J Funct Anal 255(6):1479–1553, 2008). This approach makes it possible to improve earlier well-posedness results obtained in the conservative case by a direct approach to the stochastic Schrödinger equation. In contrast to the latter, we obtain well-posedness in the full range \([1, 1 + 4/d)\) of admissible exponents in the nonlinear part (where \(d\) is the dimension of the underlying Euclidean space), i.e., in exactly the same range as in the deterministic case.     

On a discrete‐time collocation method for the nonlinear Schrödinger equation with wave operator

We consider a discrete‐time orthogonal spline collocation scheme for solving Schrödinger equation with wave operator. The scheme is proposed recently by Wang et al. (J Comput Appl Math 235 (2011), 1993–2005) and is showed to have high‐order convergence rate when a parameter θ in the scheme is not less than $\frac{1}{4}$. In this article, we show that the result can be extended to include $\theta\in(0,\frac{1}{4})$ under an assumption. Numerical example is given to justify the theoretical result. © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013     

Smoothing evolution equations and boundary control theory

We establish a relationship between the local smoothing properties of evolution equations and boundary control theory. This relationship extends to hyperbolic equations, as well as equations of the Schrödinger type.     

Exact solutions of the time fractional nonlinear Schrödinger equation with two different methods

In the present paper, exact solutions of fractional nonlinear Schrödinger equations have been derived by using two methods: Lie group analysis and invariant subspace method via Riemann‐Liouvill derivative. In the sense of Lie point symmetry analysis method, all of the symmetries of the Schrödinger equations are obtained, and these operators are applied to find corresponding solutions. In one case, we show that Schrödinger equation can be reduced to an equation that is related to the Erdelyi‐Kober functional derivative. The invariant subspace method for constructing exact solutions is presented for considered equations.     

A smoothing property of Schrödinger equations in the critical case

This paper deals with the critical case of the global smoothing estimates for the Schrödinger equation. Although such estimates fail for critical orders of weights and smoothing, it is shown that they are still valid if one works with operators with symbols vanishing on a certain set. The geometric meaning of this set is clarified in terms of the Hamiltonian flow of the Laplacian. The corresponding critical case of the limiting absorption principle for the resolvent is also established. Obtained results are extended to dispersive equations of Schrödinger type, to hyperbolic equations and to equations of other orders.     

Maximum Principle for the Regularized Schrödinger Operator

In this paper we present analogues of the maximum principle and of some parabolic inequalities for the regularized time-dependent Schrödinger operator on open manifolds using Günter derivatives. Moreover, we study the uniqueness of bounded solutions for the regularized Schrödinger–Günter problem and obtain the corresponding fundamental solution. Furthermore, we present a regularized Schrödinger kernel and prove some convergence results. Finally, we present an explicit construction for the fundamental solution to the Schrödinger–Günter problem on a class of conformally flat cylinders and tori.     

An application of the modified decomposition method for the solution of the coupled Klein–Gordon–Schrödinger equation

The modified decomposition method has been implemented for solving a coupled Klein–Gordon–Schrödinger equation. We consider a system of coupled Klein–Gordon–Schrödinger equation with appropriate initial values using the modified decomposition method. The method does not need linearization, weak nonlinearity assumptions or perturbation theory. The numerical solutions of coupled Klein–Gordon–Schrödinger equation have been represented graphically.     

Parallel Split-Step Fourier Methods for Nonlinear Schrödinger-Type Equations

The nonlinear Schrödinger equation is of tremendous interest in both theory and applications. Various regimes of pulse propagation in optical fibers are modeled by some form of the nonlinear Schrödinger equation. In this paper we introduce sequential and parallel split-step Fourier methods for numerical simulations of the nonlinear Schrödinger-type equations. The parallel methods are implemented on the Origin 2000 multiprocessor computer. Our numerical experiments have shown that these methods give accurate results and considerable speedup.