2004/11/12 by V. S. Shchesnovich, Valery S. Shchesnovich, Shchesnovich, Valery S. +3
Computer Science · Mathematics · Physics and Astronomy · #Advanced Thermodynamics and Statistical Mechanics #Chaotic Dynamics (nlin.CD) #Computational Physics (physics.comp-ph) #FOS: Physical sciences #Numerical methods for differential equations #Other Condensed Matter (cond-mat.other) #Pattern Formation and Solitons (nlin.PS) #Quantum Information and Cryptography #cond-mat.other #nlin.CD #nlin.PS #physics.comp-ph
paper · pdf · doi:10.48550/arxiv.nlin/0411033
27 pages, 8 figures; revised and extended
openalex publication_date 2004/11/12 · arxiv created 2005/09/05 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We introduce Rayleigh functional for nonlinear systems. It is defined using the energy functional and the normalization properties of the variables of variation. The key property of the Rayleigh quotient for linear systems is preserved in our definition: the extremals of the Rayleigh functional coincide with the stationary solutions of the Euler-Lagrange equation. Moreover, the second variation of the Rayleigh functional defines stability of the solution. This gives rise to a powerful numerical optimization method in the search for the energy minimizers. It is shown that the well-known imaginary time relaxation is a special case of our method. To illustrate the method we find the stationary states of Bose-Einstein condensates in various geometries. Finally, we show that the Rayleigh functional also provides a simple way to derive analytical identities satisfied by the stationary solutions of the critical nonlinear equations. We also show that the functional used in Phys. Rev. A 66, 036612 (2002) is erroneous.