Numerical renormalization group-based approach to secular perturbation theory

TL;DR

This paper introduces a numerical renormalization group approach based on differential geometry to improve secular perturbation theory, enabling long-term solutions for systems where analytic solutions are unavailable.

Contribution

It reformulates the dynamical renormalization group in a geometric framework, allowing application to numerical solutions and extending perturbation analysis to any order.

Findings

Successfully applied to the Korteweg-de Vries equation with damping

Constructed long-term solutions beyond naive perturbation theory

Extended DRG applicability to systems with parameter flows

Abstract

Perturbation theory is a crucial tool for many physical systems, when exact solutions are not available, or nonperturbative numerical solutions are intractable. Naive perturbation theory often fails on long timescales, leading to secularly growing solutions. These divergences have been treated with a variety of techniques, including the powerful dynamical renormalization group (DRG). Most of the existing DRG approaches rely on having analytic solutions up to some order in perturbation theory. However, sometimes the equations can only be solved numerically. We reformulate the DRG in the language of differential geometry, which allows us to apply it to numerical solutions of the background and perturbation equations. This formulation also enables us to use the DRG in systems with background parameter flows, and therefore, extend our results to any order in perturbation theory. As an example, we apply this method to calculate the soliton-like solutions of the Korteweg-de Vries equation deformed by adding a small damping term. We numerically construct DRG solutions which are valid on secular time scales, long after naive perturbation theory has broken down.