Spectral approach to Korteweg-de Vries equations on the compactified real line

TL;DR

This paper introduces a spectral numerical method for solving generalized Korteweg-de Vries equations on the real line by compactifying the domain and employing Chebyshev collocation and implicit Runge-Kutta time integration.

Contribution

It presents a novel spectral approach combining domain compactification and Chebyshev collocation for KdV equations on an unbounded domain.

Findings

Effective handling of non-vanishing initial data at infinity.

Applicable to data with slow decay, not satisfying Faddeev condition.

Demonstrated accuracy and stability of the numerical method.

Abstract

We present a numerical approach for generalised Korteweg-de Vries (KdV) equations on the real line. In the spatial dimension we compactify the real line and apply a Chebyshev collocation method. The time integration is performed with an implicit Runge-Kutta method of fourth order. Several examples are discussed: initial data bounded but not vanishing at infinity as well as data not satisfying the Faddeev condition, i.e. with a slow decay towards infinity.