From Klein-Gordon-Wave to Schr\"odinger-Wave: a Normal Form Approach

In this study, we consider a Klein-Gordon-Wave system, which couples the evolution of a massive field and a massless one through a Yukawa interaction and we derive its Hamiltonian normal form to second order. To the first-order approximation, the normal form results in a Schr\"odinger-Wave system, while a Schr\"odinger-Poisson system is derived in the limit of vanishing perturbative parameter. A second-order approximation provides the successive corrections to the Schr\"odinger-Wave system, while higher-order approximations can be obtained by iterating our constructive procedure.