Carleman Linearization of Partial Differential Equations

TL;DR

This paper extends Carleman linearization to partial differential equations with quadratic nonlinearities, demonstrating its application on Burger's and Vlasov equations to facilitate linear analysis of complex PDE systems.

Contribution

The authors generalize Carleman linearization to PDEs with quadratic nonlinearities while preserving the original structure, enabling new analytical approaches.

Findings

Successfully applied to Burger's equation

Extended to Vlasov equation

Maintains original structure of Carleman linearization

Abstract

Carleman linearization is a technique that embeds systems of ordinary differential equations with polynomial nonlinearities into infinite dimensional linear systems in a procedural way. In this paper we generalize the method for systems of partial differential equations with quadratic nonlinearities, while maintaining the original structure of Carleman linearization. Furthermore, we apply our approach to Burger's equation and to the Vlasov equation as examples.