Conjugate gradient method for finding fundamental solitary waves

TL;DR

This paper introduces a modified conjugate gradient method tailored for efficiently finding fundamental solitary waves in nonlinear Hamiltonian equations, overcoming spectral challenges and outperforming existing methods in convergence speed.

Contribution

The authors develop a novel modification of the conjugate gradient method capable of finding solitary waves with specific parameters in nonlinear Hamiltonian systems, including multi-component cases.

Findings

Modified CGM converges faster than Petviashvili's method.

Effective for solitary waves with prescribed parameters.

Applicable to multi-component nonlinear wave equations.

Abstract

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.