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

TL;DR

This paper introduces a mode elimination technique that enhances the convergence of iterative methods for finding solitary waves by removing problematic modes, enabling faster convergence and access to nonfundamental solutions.

Contribution

It proposes a novel mode elimination approach that improves convergence and allows finding nonfundamental solitary waves, extending the generalized Petviashvili method.

Findings

The mode elimination technique accelerates convergence of iterative methods.

It enables the computation of nonfundamental solitary waves.

Comparison of iteration operators explains differences in convergence rates.

Abstract

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.