Lax Connection and Conserved Quantities of Quadratic Mean Field Games

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.