2009/03/19 by Taras I. Lakoba · 2 citations
Engineering · Mathematics · Physics and Astronomy · #Applied mathematics #Computer science #Conjugate gradient method #Constant (computer programming) #Convergence (economics) #Eigenvalues and eigenvectors #Iterative method #Mathematical analysis #Mathematical optimization #Mathematics #Nonlinear Photonic Systems #Nonlinear Waves and Solitons #Nonlinear system #Operator (biology) #Optical Network Technologies #Physics #Power iteration #Sign (mathematics) #nlin.PS
paper · pdf · doi:10.1016/j.physd.2009.09.013
44 pages, submitted to Physica D
arxiv created 2009/03/19 · openalex publication_date 2009/09/20 · arxiv updated 2015/05/13 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
The Conjugate Gradient method (CGM) is known to be the fastest generic iterative method for solving linear systems with symmetric sign definite matrices. In this paper, we modify this method so that it could find fundamental solitary waves of nonlinear Hamiltonian equations. The main obstacle that such a modified CGM overcomes is that the operator of the equation linearized about a solitary wave is not sign definite. Instead, it has a finite number of eigenvalues on the opposite side of zero than the rest of its spectrum. We present versions of the modified CGM that can find solitary waves with prescribed values of either the propagation constant or power. We also extend these methods to handle multi-component nonlinear wave equations. Convergence conditions of the proposed methods are given, and their practical implications are discussed. We demonstrate that our modified CGMs converge much faster than, say, Petviashvili's or similar methods, especially when the latter converge slowly.