Energy minimization related to the nonlinear Schrödinger equation

DOI10.1016/j.jcp.2008.12.016zbMath1161.65339OpenAlexW2002704696WikidataQ62699600 ScholiaQ62699600MaRDI QIDQ1008872

Shahid S. Siddiqi, Sultan Sial, Nauman Raza, Turab Lookman

Publication date: 30 March 2009

Published in: Journal of Computational Physics (Search for Journal in Brave)

Full work available at URL: https://doi.org/10.1016/j.jcp.2008.12.016

zbMATH Keywords

numerical examplesnonlinear Schrödinger equationfinite-differencefinite-elementSchrödinger functionalSobolev gradientsminimum energy states

Mathematics Subject Classification ID

Numerical optimization and variational techniques (65K10) Newton-type methods (49M15) Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics (81Q05)

Related Items (14)

Gradient flow finite element discretizations with energy-based adaptivity for the Gross-Pitaevskii equation ⋮ Numerical approximation of time evolution related to Ginzburg-Landau functionals using weighted Sobolev gradients ⋮ A finite element method with mesh adaptivity for computing vortex states in fast-rotating Bose-Einstein condensates ⋮ Gradient Flow Finite Element Discretisations with Energy-Based Adaptivity for Excited States of Schrödinger's Equation ⋮ An overview of \textit{a posteriori} error estimation and post-processing methods for nonlinear eigenvalue problems ⋮ Sobolev Gradient Flow for the Gross--Pitaevskii Eigenvalue Problem: Global Convergence and Computational Efficiency ⋮ Numerical solution of Burgers equation by the Sobolev gradient method ⋮ Numerical solutions of singularly perturbed reaction diffusion equation with Sobolev gradients ⋮ Application of Sobolev gradient method to Poisson-Boltzmann system ⋮ The \(J\)-method for the Gross-Pitaevskii eigenvalue problem ⋮ Finite-time stability and stabilization of nonlinear stochastic hybrid systems ⋮ MINIMIZING THE GROSS-PITAEVSKII ENERGY FUNCTIONAL WITH THE SOBOLEV GRADIENT — ANALYTICAL AND NUMERICAL RESULTS ⋮ An accelerated Sobolev gradient method for unconstrained optimization problems based on variable inner products ⋮ Polynomial solution of singular differential equations using Weighted Sobolev gradients

Uses Software

• FreeFem++

Cites Work

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• On the recovery of transport parameters in groundwater modelling
• Preconditioning operators and Sobolev gradients for nonlinear elliptic problems
• Sobolev gradient preconditioning for the electrostatic potential equation
• Energy minimization using Sobolev gradients: application to phase separation and ordering.
• A quasi-local Gross-Pitaevskii equation for attractive Bose-Einstein condensates
• Optimizing Schrödinger Functionals Using Sobolev Gradients: Applications to Quantum Mechanics and Nonlinear Optics
• A variational approach to an elastic inverse problem
• Steepest descent using smoothed gradients

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