On removing orders from amplitude equations

TL;DR

This paper presents a modified renormalization group method that simplifies amplitude equations by removing certain terms, improving accuracy while avoiding linear growth reintroduction, with tests on scalar and system ODEs.

Contribution

The paper introduces a new homogeneous function in the RG method to selectively remove terms from amplitude equations, revealing a limit to term removal and maintaining accuracy.

Findings

Effective removal of terms up to a limit without reintroducing linear growth

Enhanced accuracy in amplitude equations with unchanged complexity

Validated on various scalar and system ODEs

Abstract

In this paper, we introduce a modified version of the renormalization group (RG) method and test its numerical accuracy. It has been tested on numerous scalar ODEs and systems of ODEs. Our method is primarily motivated by the possibility of simplifying amplitude equations. The key feature of our method is the introduction of a new homogeneous function at each order of the perturbation hierarchy, which is then used to remove terms from the amplitude equations. We have shown that there is a limit to how many terms can be removed, as doing so beyond a certain point would reintroduce linear growth. There is thus a \textit{core} in the amplitude equation, which consists of the terms that cannot be removed while avoiding linear growth. Using our modified RG method, higher accuracy can also be achieved while maintaining the same level of complexity in the amplitude equation.