Optimizing Schrödinger Functionals Using Sobolev Gradients: Applications to Quantum Mechanics and Nonlinear Optics

Authors:
Juan José García-Ripoll;Víctor M. Pérez-García
Affiliations:
-;-
Venue:
SIAM Journal on Scientific Computing
Year:
2001

Citing 0
Cited 13

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A finite element method with mesh adaptivity for computing vortex states in fast-rotating Bose-Einstein condensates

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A spectral-Galerkin continuation method using Chebyshev polynomials for the numerical solutions of the Gross-Pitaevskii equation

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A New Sobolev Gradient Method for Direct Minimization of the Gross-Pitaevskii Energy with Rotation

SIAM Journal on Scientific Computing
Original article: Convergence conditions for iterative methods seeking multi-component solitary waves with prescribed quadratic conserved quantities

Mathematics and Computers in Simulation
Spectral collocation methods using sine functions for a rotating Bose-Einstein condensation in optical lattices

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Abstract

In this paper we study the application of the Sobolev gradients technique to the problem of minimizing several Schrödinger functionals related to timely and difficult nonlinear problems in quantum mechanics and nonlinear optics. We show that these gradients act as preconditioners over traditional choices of descent directions in minimization methods and show a computationally inexpensive way to obtain them using a discrete Fourier basis and a fast Fourier transform. We show that the Sobolev preconditioning provides a great convergence improvement over traditional techniques for finding solutions with minimal energy as well as stationary states and suggest a generalization of the method using arbitrary linear operators.