Simulation of coherent structures in nonlinear Schrödinger-type equations

Authors:
I. Alonso-Mallo;A. Durán;N. Reguera
Affiliations:
Departamento de Matemática Aplicada, Facultad de Ciencias, Universidad de Valladolid, C/Dr. Mergelina s/n, 47011 Valladolid, Spain;Departamento de Matemática Aplicada, E.T.S.I. Telecomunicación, Campus Miguel Delibes, Universidad de Valladolid, Paseo de Belen 15, 47011 Valladolid, Spain;Departamento de Matemáticas y Computación, Escuela Politécnica Superior, Universidad de Burgos, Avda. de Cantabria s/n, 09006 Burgos, Spain
Venue:
Journal of Computational Physics
Year:
2010

Citing 26
Cited 0

Split-step methods for the solution of the nonlinear Schro¨dinger equation

SIAM Journal on Numerical Analysis
Generalizations of Arakawa's Jacobian

Journal of Computational Physics
Long time behavior of solutions of nonlinear classical field equations: the example of NLS defocusing

Proceedings of a NATO advanced research workshop on New trends in nonlinear dynamics : nonvariational aspects: nonvariational aspects
Numerical simulation of nonlinear Schro¨dinger systems: a new conservative scheme

Applied Mathematics and Computation
A mean-field statistical theory for the nonlinear Schrödinger equation

Physica D
Statistical equilibrium states for the nonlinear Schrödinger equation

Mathematics and Computers in Simulation - IMACS sponsored special issue on nonlinear waves: computation and theory
Dissipative or conservative finite-difference schemes for complex-valued nonlinear partial differential equations

Journal of Computational Physics
Self-focusing on bounded domains

Physica D
Finite-difference schemes for nonlinear wave equation that inherit energy conservation property

Journal of Computational and Applied Mathematics
On time-splitting spectral approximations for the Schrödinger equation in the semiclassical regime

Journal of Computational Physics
Linearly implicit methods for the nonlinear Schrödinger equation in nonhomogeneous media

Applied Mathematics and Computation
Ground-state solution of Bose--Einstein condensate by directly minimizing the energy functional

Journal of Computational Physics
Numerical solution of the Gross--Pitaevskii equation for Bose--Einstein condensation

Journal of Computational Physics
Computing the Ground State Solution of Bose--Einstein Condensates by a Normalized Gradient Flow

SIAM Journal on Scientific Computing
Convergence of Petviashvili's Iteration Method for Numerical Approximation of Stationary Solutions of Nonlinear Wave Equations

SIAM Journal on Numerical Analysis
Fourth-Order Time-Stepping for Stiff PDEs

SIAM Journal on Scientific Computing
A Fourth-Order Time-Splitting Laguerre--Hermite Pseudospectral Method for Bose--Einstein Condensates

SIAM Journal on Scientific Computing
Exact nonreflecting boundary conditions for one-dimensional cubic nonlinear Schrödinger equations

Journal of Computational Physics
A high order finite element discretization with local absorbing boundary conditions of the linear Schrödinger equation

Journal of Computational Physics
Why Newton's method is hard for travelling waves: Small denominators, KAM theory, Arnold's linear Fourier problem, non-uniqueness, constraints and erratic failure

Mathematics and Computers in Simulation
A generalized Petviashvili iteration method for scalar and vector Hamiltonian equations with arbitrary form of nonlinearity

Journal of Computational Physics
A mode elimination technique to improve convergence of iteration methods for finding solitary waves

Journal of Computational Physics
Time behaviour of the error when simulating finite-band periodic waves. The case of the KdV equation

Journal of Computational Physics
Symmetric Exponential Integrators with an Application to the Cubic Schrödinger Equation

Foundations of Computational Mathematics
A minimisation approach for computing the ground state of Gross-Pitaevskii systems

Journal of Computational Physics
High-order time-splitting Hermite and Fourier spectral methods

Journal of Computational Physics

Quantified Score

Hi-index	31.45

Visualization

Abstract

This paper presents some numerical methods to simulate the evolution of coherent structures with small fluctuations, that appear as typical solutions of a class of nonintegrable nonlinear Schrodinger equations. The construction of the methods is particularly focused on two points: on one hand, the generation of the ground state profiles, to be used in the initial data of the simulations, combines a suitable spatial discretization with the resolution of a discrete variational problem. On the other hand, the approximation to leading parameters of these structures is controlled by the time integration. We compare different methods when simulating the evolution of initial ground state profiles and some initial data perturbed from them.