Two solvable systems of coagulation equations with limited aggregations

TL;DR

This paper introduces two solvable models of polymer formation with limited aggregations, providing explicit solutions to modified coagulation equations and analyzing their asymptotic behavior, connecting to known Smoluchowski models.

Contribution

It presents new exactly solvable coagulation models with aggregation limits, solving the associated nonlinear equations explicitly and exploring their long-term behavior.

Findings

Explicit solutions to the modified coagulation equations.

Asymptotic analysis of the solutions.

Connections to classical Smoluchowski models.

Abstract

We consider two simple models for the formation of polymers where at the initial time, each monomer has a certain number of potential links (called arms in the text) that are consumed when aggregations occur. Loosely speaking, this imposes restrictions on the number of aggregations. The dynamics of concentrations are governed by modifications of Smoluchowski's coagulation equations. Applying classical techniques based on generating functions, resolution of quasi-linear PDE's, and Lagrange inversion formula, we obtain explicit solutions to these non-linear systems of ODE's. We also discuss the asymptotic behavior of the solutions and point at some connexions with certain known solutions to Smoluchowski's coagulation equations with additive or multiplicative kernels.