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

We extend the key idea behind the generalized Petviashvili method of Ref. \cite{gP} by proposing a novel technique based on a similar idea. This technique systematically eliminates from the iteratively obtained solution a mode that is "responsible" either for the divergence or the slow convergence of the iterations. We demonstrate, theoretically and with examples, that this mode elimination technique can be used both to obtain some nonfundamental solitary waves and to considerably accelerate convergence of various iteration methods. As a collateral result, we compare the linearized iteration operators for the generalized Petviashvili method and the well-known imaginary-time evolution method and explain how their different structures account for the differences in the convergence rates of these two methods.Comment: to appear in J. Comp. Phys.; 24 page

Conjugate gradient method for finding fundamental solitary waves

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.Comment: 44 pages, submitted to Physica

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

The Petviashvili's iteration method has been known as a rapidly converging numerical algorithm for obtaining fundamental solitary wave solutions of stationary scalar nonlinear wave equations with power-law nonlinearity: \ $-Mu+u^p=0$, where $M$ is a positive definite self-adjoint operator and $p={\rm const}$. In this paper, we propose a systematic generalization of this method to both scalar and vector Hamiltonian equations with arbitrary form of nonlinearity and potential functions. For scalar equations, our generalized method requires only slightly more computational effort than the original Petviashvili method.Comment: to appear in J. Comp. Phys.; 35 page