Asymptotic Analysis of a bi-monomeric nonlinear Becker-D{\"o}ring system

TL;DR

This paper analyzes a bi-monomeric nonlinear Becker-Döring system to explain sustained and damped oscillations in depolymerization, characterizing phases, periods, damping, and providing numerical methods.

Contribution

It offers a detailed asymptotic analysis of oscillatory behavior and damping in a bi-monomeric Becker-Döring model, with quantitative approximations and numerical validation.

Findings

Characterization of oscillation periods and damping phases.

Quantitative estimates of oscillation amplitude and energy loss.

Numerical methods for solving the system effectively.

Abstract

To provide a mechanistic explanation of sustained {then} damped oscillations observed in a depolymerisation experiment, a bi-monomeric variant of the seminal Becker-D\"oring system has been proposed in \cite{DFMR}. When all reaction rates are constant, the equations are the following: \begin{align*}\frac{dv}{dt} & =-vw+v\sum_{j=2}^{\infty}c_{j}, \qquad \frac{dw}{dt} =vw-w\sum_{j=1}^{\infty}c_{j}, \\ \frac{dc_{j}}{dt} & =J_{j-1}-J_{j}\ \ ,\ \ j\geq1\ \ ,\ \ \ J_{j}=wc_{j}-vc_{j+1}\ \ ,\ \ j\geq1\ \ ,\ J_{0}=0, \end{align*} where $v$ and $w$ are two distinct unit species, and $c_i$ represents the concentration of clusters containing $i$ units. We study in detail the mechanisms leading to such oscillations and characterise the different phases of the dynamics, from the initial high-amplitude oscillations to the progressive damping leading to the convergence towards the unique positive stationary solution. We give quantitative approximations for the main quantities of interest: period of the oscillations, size of the damping (corresponding to a loss of energy), number of oscillations characterising each phase. We illustrate these results by numerical simulation, in line with the theoretical results, and provide numerical methods to solve the system.