2,494 research outputs found
A family of high-order multistep methods with vanished phase-lag and its derivatives for the numerical solution of the Schrödinger equation
AbstractMany simulation algorithms (chemical reaction systems, differential systems arising from the modelling of transient behaviour in the process industries etc.) contain the numerical solution of systems of differential equations. For the efficient solution of the above mentioned problems, linear multistep methods or Runge–Kutta single-step methods are used. For the simulation of chemical procedures the radial Schrödinger equation is used frequently. In the present paper we will study a class of linear multistep methods. More specifically, the purpose of this paper is to develop an efficient algorithm for the approximate solution of the radial Schrödinger equation and related problems. This algorithm belongs in the category of the multistep methods. In order to produce an efficient multistep method the phase-lag property and its derivatives are used. Hence the main result of this paper is the development of an efficient multistep method for the numerical solution of systems of ordinary differential equations with oscillating or periodical solutions. The reason of their efficiency, as the analysis proved, is that the phase-lag and its derivatives are eliminated. Another reason of the efficiency of the new obtained methods is that they have high algebraic orde
Nearly chirp- and pedestal-free pulse compression in nonlinear fiber Bragg gratings
Peer reviewedPublisher PD
New modified Runge–Kutta–Nyström methods for the numerical integration of the Schrödinger equation
AbstractIn this work we construct new Runge–Kutta–Nyström methods especially designed to integrate exactly the test equation y″=−w2y. We modify two existing methods: the Runge–Kutta–Nyström methods of fifth and sixth order. We apply the new methods to the computation of the eigenvalues of the Schrödinger equation with different potentials such as the harmonic oscillator, the doubly anharmonic oscillator and the exponential potential
An algebraic method to solve the radial Schrödinger equation
AbstractWe propose a method of numerical integration of differential equations of the type x2y″+f(x)y=0 by approximating its solution with solutions of equations of the type x2y″+(ax2+bx+c)y=0. This approximation is performed by segmentary approximation on an interval. We apply the method to obtain approximate solutions of the radial Schrödinger equation on a given interval and test it for two different potentials. We conclude that our method gives a similar accuracy than the Taylor method of higher order
Oscillons and oscillating kinks in the Abelian-Higgs model
We study the classical dynamics of the Abelian Higgs model employing an
asymptotic multiscale expansion method, which uses the ratio of the Higgs to
the gauge field amplitudes as a small parameter. We derive an effective
nonlinear Schr\"{o}dinger equation for the gauge field, and a linear equation
for the scalar field containing the gauge field as a nonlinear source. This
equation is used to predict the existence of oscillons and oscillating kinks
for certain regimes of the ratio of the Higgs to the gauge field masses.
Results of numerical simulations are found to be in very good agreement with
the analytical findings, and show that the oscillons are robust, while kinks
are unstable. It is also demonstrated that oscillons emerge spontaneously as a
result of the onset of the modulational instability of plane wave solutions of
the model. Connections of the obtained solutions with the phenomenology of
superconductors is discussed.Comment: arXiv admin note: substantial text overlap with arXiv:1306.386
Interaction instability of localization in quasiperiodic systems
Integrable models form pillars of theoretical physics because they allow for
full analytical understanding. Despite being rare, many realistic systems can
be described by models that are close to integrable. Therefore, an important
question is how small perturbations influence the behavior of solvable models.
This is particularly true for many-body interacting quantum systems where no
general theorems about their stability are known. Here, we show that no such
theorem can exist by providing an explicit example of a one-dimensional
many-body system in a quasiperiodic potential whose transport properties
discontinuously change from localization to diffusion upon switching on
interaction. This demonstrates an inherent instability of a possible many-body
localization in a quasiperiodic potential at small interactions. We also show
how the transport properties can be strongly modified by engineering potential
at only a few lattice sites.Comment: 10 pages; (v2: additional explanations, data, and references
Singular Short Range Potentials in the J-Matrix Approach
We use the tools of the J-matrix method to evaluate the S-matrix and then
deduce the bound and resonance states energies for singular screened Coulomb
potentials, both analytic and piecewise differentiable. The J-matrix approach
allows us to absorb the 1/r singularity of the potential in the reference
Hamiltonian, which is then handled analytically. The calculation is performed
using an infinite square integrable basis that supports a tridiagonal matrix
representation for the reference Hamiltonian. The remaining part of the
potential, which is bound and regular everywhere, is treated by an efficient
numerical scheme in a suitable basis using Gauss quadrature approximation. To
exhibit the power of our approach we have considered the most delicate region
close to the bound-unbound transition and compared our results favorably with
available numerical data.Comment: 14 pages, 5 tables, 2 figure
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