Discrete Schrodinger Equation Research Articles

On solitary wave solutions of a peptide group system with higher order saturable nonlinearity

Abstract In this article, a new continuous model of a peptide group system with higher order saturable nonlinearity is derived. The model is constructed up on discrete Schrodinger equation with higher order saturable nonlinearity. New exact solutions of the proposed model are investigated. Via the generalized Riccati equation mapping method travelling wave and soliton solutions are derived. For certain values of parameters kink, antikink, breathers, and dark and bright solitons are presented.

Even-odd-dependent optical transitions of zigzag monolayer black phosphorus nanoribbons

We analytically study the electronic structure and optical properties of zigzag-edged phosphorene nanoribbons (ZPNRs) using the tight-binding Hamiltonian and Kubo formula. By directly solving the discrete Schrodinger equation, we obtain the energy spectra and wavefunctions for N-ZPNR (where TV is the number of transverse zigzag atomic chains) and classify the eigenstates according to the lattice symmetry. Then, we obtain the optical transition selection rule of ZPNRs on the basis of symmetry analysis and analytical expressions of optical transition matrix elements. Under incident light that is linearly polarized along the ribbon, we determine that the optical transition selection rule for N-ZPNR with even- or odd-N is qualitatively different. Specifically, for even-N ZPNRs, the inter- (intra-) band selection rule is An =odd (even) because the parity of the wavefunction corresponding to the n-th subband in the conduction (valence) band is (-1)n[(-1)(n+1)] owing to the presence of C2x symmetry. However, the optical transitions between any subbands are possible owing to the absence of C2x symmetry. Our results provide a further understanding on the electronic states and optical properties of ZPNRs, which are useful for explaining the optical experiment data on ZPNR samples.

Local Perturbation of the Discrete Schrödinger Operator and a Generalized Chebyshev Oscillator

We discuss the conditions under which a special linear transformation of the classical Chebyshev polynomials (of the second kind) generate a class of polynomials related to “local perturbations” of the coefficients of a discrete Schrodinger equation. These polynomials are called generalized Chebyshev polynomials. We answer this question for the simplest class of “local perturbations” and describe a generalized Chebyshev oscillator corresponding to generalized Chebyshev polynomials.

Convexity properties of discrete Schrödinger evolutions

In this paper, we give log-convexity properties for solutions to discrete Schrodinger equations with different discrete versions of Gaussian decay at two different times. For free evolutions, we use complex analysis arguments to derive these properties, while in a perturbative setting we use a preliminar log-convexity statement in order to conclude our result.

A discrete Hardy’s uncertainty principle and discrete evolutions

In this paper we give a discrete version of Hardy’s uncertainty principle, by using complex variable arguments, as in the classical proof of Hardy’s principle. Moreover, we give an interpretation of this principle in terms of decaying solutions to the discrete Schrodinger and heat equations.

Wannier functions and discrete NLS equations for nematicons$^\dagger$

We derive nonlocal discrete nonlinear Schrodinger (DNLS) equations for laser beam propagation in optical waveguide arrays that use a nematic liquid crystal substrate. We start with an NLS-elliptic model for the problem and propose a simplified version that incorporates periodicity in one of the directions transverse to the propagation of the beam. We use Wannier basis functions for an associated Schrodinger operator with periodic potential to derive discrete equations for Wannier modes and propose some possible simplified systems for interactions of modes within the first energy band of the periodic Schrodinger operator. In particular, we present the simplest generalization of a model proposed by Fratalocchi and Assanto by including a linear nonlocal term, and see evidence for parameter regimes where nonlinearity is more pronounced.

Multiple solutions of discrete Schrödinger equations with growing potentials

Under some weaker conditions than elsewhere, we obtain infinitely many homoclinic solutions for a class of discrete Schrodinger equations in infinite m dimensional lattices with nonlinearities being superlinear at infinity by using variational methods. Our result extends some existing results in the literature.

Non-periodic discrete Schrödinger equations: ground state solutions

In this paper, we study a class of non-periodic discrete Schrodinger equations with superlinear non-linearities at infinity. Under conditions weaker than those previously assumed, we obtain the existence of ground state solutions, i.e., non-trivial solutions with least possible energy. In addition, an example is given to illustrate our results.

Transient super-ballistic spreading of wave packets with large spreading exponents in some hybrid ordered-quasiperiodic lattices

We consider the spreading of an initially localized wave packet in one-dimensional hybrid ordered-quasiperiodic lattices. We consider two diffrent kinds of quasiperiodic sequences, which are the Cantor and the period-doubling sequences. From numerical calculations based on the discrete Schrodinger equation, we demonstrate that hybrid ordered-quasiperiodic lattices can support the super-ballistic spreading of a wave packet with very large spreading exponents for certain transient time windows. Remarkably, in the case of the sublattice with the on-site potential obeying the period-doubling quasiperiodic sequence, we find that the super-ballistic exponent can be larger than six. We also point out that previous explanations of this phenomenon based on a generalized version of the point source model are incorrect.

Quasi-bounded supersolutions of discrete Schrodinger equations

Quasi-bounded supersolutions of discrete Schrodinger equations

Antibound state for the discrete Schrodinger equation

Antibound state for the discrete Schrodinger equation

Numerical Solution of Fractional Partial Differential Equations by Discrete Adomian Decomposition Method

AbstractIn this paper we find the solution of linear as well as nonlinear fractional partial differential equations using discrete Adomian decomposition method. Here we develop the discrete Adomian decomposition method to find the solution of fractional discrete diffusion equation, nonlinear fractional discrete Schrodinger equation, fractional discrete Ablowitz-Ladik equation and nonlinear fractional discrete Burger’s equation. The obtained solution is verified by comparison with exact solution whenα= 1.

A Generalized FDTD Method with Absorbing Boundary Condition for Solving a Time-Dependent Linear Schrodinger Equation

The Finite-Difference Time-Domain (FDTD) method is a well-known technique for the analysis of quantum devices. It solves a discretized Schrodinger equation in an iterative process. However, the method provides only a second-order accurate numerical solution and requires that the spatial grid size and time step should satisfy a very restricted condition in order to prevent the numerical solution from diverging. In this article, we present a generalized FDTD method with absorbing boundary condition for solving the one-dimensional (1D) time-dependent Schr?dinger equation and obtain a more relaxed condition for stability. The generalized FDTD scheme is tested by simulating a particle moving in free space and then hitting an energy potential. Numerical results coincide with those obtained based on the theoretical analysis.

Discrete solitons in electromechanical resonators

We consider a particular type of parametrically driven discrete Klein-Gordon system describing microdevices and nanodevices, with integrated electrical and mechanical functionality. Using a multiscale expansion method we reduce the system to a discrete nonlinear Schrödinger equation. Analytical and numerical calculations are performed to determine the existence and stability of fundamental bright and dark discrete solitons admitted by the Klein-Gordon system through the discrete Schrödinger equation. We show that a parametric driving can not only destabilize onsite bright solitons, but also stabilize intersite bright discrete solitons and onsite and intersite dark solitons. Most importantly, we show that there is a range of values of the driving coefficient for which dark solitons are stable, for any value of the coupling constant, i.e., oscillatory instabilities are totally suppressed. Stability windows of all the fundamental solitons are presented and approximations to the onset of instability are derived using perturbation theory, with accompanying numerical results. Numerical integrations of the Klein-Gordon equation are performed, confirming the relevance of our analysis.

Soliton stability and collapse in the discrete nonpolynomial Schrödinger equation with dipole-dipole interactions

The stability and collapse of fundamental unstaggered bright solitons in the discrete Schrodinger equation with the nonpolynomial on-site nonlinearity, which models a nearly one-dimensional Bose-Einstein condensate trapped in a deep optical lattice, are studied in the presence of the long-range dipole-dipole (DD) interactions. The cases of both attractive and repulsive contact and DD interaction are considered. The results are summarized in the form of stability/collapse diagrams in the parametric space of the model, which demonstrate that the the attractive DD interactions stabilize the solitons and help to prevent the collapse. Mobility of the discrete solitons is briefly considered too.

Boundary observability of time discrete Schrodinger equations

In this paper we study the boundary observability estimate of time discrete Schrodinger equations in a bounded domain. By means of a time discrete version of the classical multiplier technique, we prove the uniform observability inequality of the solutions in an appropriate filtered space in which the high frequency components have been filtered. In this way, the well-known boundary observability property of the Schodinger equation can be reproduced as the limit, as , h → 0 of the observability of the time discrete one. Better than the existing result in Ervedoza et al. (2008), our alterative proof shows the rigorous relationship between the filtering parameter and the optimal observation time T. Moreover, the latter one tends to zero as the time scale tends to zero. Finally, the optimality of the order of the filtering parameter is also established for lower dimensional case.

Existence of solitary waves for the discrete Schrödinger equation coupled to a nonlinear oscillator

The discrete Schrodinger equation with a nonlinearity concentrated at a single point is an interesting and important model to study the long-time behavior of solutions, including the asymptotic stability of solitary waves and properties of global attractors. In this note, the global well-posedness of this equation and the existence of solitary waves is proved and the properties of these waves are studied.

Singularity avoidance, lattices, and quantum gravity

An ingredient in recent discussions of curvature singularity avoidance in quantum gravity is the “inverse scale factor” operator in quantum cosmology, and its generalizations to field theoretic models such as scalar-field collapse in spherical symmetry. I describe a general lattice origin of this idea, and show how it applies to the Coulomb singularity in quantum mechanics. The example demonstrates that a discretized Schrodinger equation is computationally equivalent to the so-called polymer quantization derived loop quantum gravity. This applies also to lattice discretized forms of the Wheeler–deWitt equation.PACS Nos.: 04.60.–m, 04.60.Ds, 04.70.Dy

Method of Variations of Potential of Quasi-Periodic Schrödinger Equations

We study the one-dimensional discrete quasi-periodic Schrodinger equation $$-{\varphi}(n + 1) -{\varphi}(n - 1) + {\lambda}V (x + n\omega){\varphi}(n) = E{\varphi}(n),\quad n \in {\mathbb{Z}}$$ . We introduce the notion of variations of potential and use it to define typical potential. We show that for typical C3 potential V, if the coupling constant λ is large, then for most frequencies ω, the Lyapunov exponent is positive for all energies E.

Inverse problem for the two-dimensional discrete Schrödinger equation in a square

The potential of the two-dimensional discrete Schrodinger equation can be reconstructed from a part of its spectrum and prescribed symmetry conditions of the basis eigenfunctions. The discrete potential along with the missing eigenvalues is found by solving a polynomial system of equations, which is derived and solved using the REDUCE computer algebra system. To ensure the convergence of the iterative process implemented in the Numeric package in REDUCE, proper initial data must be specified. The prescribed eigenvalues are perturbed original eigenvalues corresponding to the zero discrete potential. The original eigenvalues provide the natural initial data for the corresponding missing eigenvalues. In the case of a square, there are many multiple eigenvalues among the original eigenvalues. The direct application of the variant of the Newton method implemented in the Numeric package in REDUCE is impossible in the case of multiple initial data. A modification of the method proposed earlier for calculating the discrete potential of the two-dimensional discrete Schrodinger equation in a square is illustrated by an example.