Multi-domain spectral approach for the Hilbert transform on the real line

TL;DR

This paper introduces a multi-domain spectral method for computing the Hilbert transform on the entire real line, especially for functions with specific decay or analyticity properties, and applies it to soliton solutions of generalized Benjamin-Ono equations.

Contribution

It presents a novel spectral approach for the Hilbert transform on the real line, accommodating piece-wise analytic and algebraically decaying functions, with applications to soliton construction.

Findings

Effective computation of the Hilbert transform for various function types.

Successful construction of solitons for generalized Benjamin-Ono equations.

Demonstrated accuracy and applicability of the method.

Abstract

A multi-domain spectral method is presented to compute the Hilbert transform on the whole compactified real line, with a special focus on piece-wise analytic functions and functions with algebraic decay towards infinity. Several examples of these and other types of functions are discussed. As an application solitons to generalized Benjamin-Ono equations are constructed.