Lax Connection and Conserved Quantities of Quadratic Mean Field Games

TL;DR

This paper explores the integrability of quadratic mean field game systems, revealing a connection to nonlinear Schrödinger equations and identifying conserved quantities that deepen understanding of their solution structures.

Contribution

It establishes a link between mean field game systems and nonlinear Schrödinger equations, demonstrating integrability and conserved quantities in certain cases.

Findings

Existence of a hierarchy of conserved quantities.

Connection between mean field games and nonlinear Schrödinger equations.

New questions on integrability in mean field game systems.

Abstract

Mean Field Game is a rather new field initially developed in applied mathematics and engineering in order to deal with the dynamics of a large number of controlled agents or objects in interaction. For a large class of these models, there exists a deep relationship between the associated system of equations and the non linear Schr\"odinger equation, which allows to get new insights on the structure of their solutions. In this work, we deal with related aspects of integrability for such systems, exhibiting in some cases a full hierarchy of conserved quantities, and bringing some new questions which arise in this specific context.