Dynamics of the nonlinear Klein–Gordon equation in the nonrelativistic limit

  • S. PasqualiEmail author


We study the nonlinear Klein–Gordon (NLKG) equation on a manifold M in the nonrelativistic limit, namely as the speed of light c tends to infinity. In particular, we consider a higher-order normalized approximation of NLKG (which corresponds to the NLS at order \(r=1\)) and prove that when M is a smooth compact manifold or \(\mathbb {R}^d\), the solution of the approximating equation approximates the solution of the NLKG locally uniformly in time. When \(M=\mathbb {R}^d\), \(d \ge 2\), we also prove that for \(r \ge 2\) small radiation solutions of the order-r normalized equation approximate solutions of the nonlinear NLKG up to times of order \(\mathscr {O}(c^{2(r-1)})\). We also prove a global existence result uniform with respect to c for the NLKG equation on \(\mathbb {R}^3\) with cubic nonlinearity for small initial data and Strichartz estimates for the Klein–Gordon equation with potential on \(\mathbb {R}^3\).


Nonrelativistic limit Nonlinear Klein–Gordon equation Birkhoff normal form Long-time behavior 

Mathematics Subject Classification

37K55 70H08 70K45 81Q05 



The author would like to thank Professor Dario Bambusi, for introducing him to the problem and for many valuable discussions and suggestions.


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© Fondazione Annali di Matematica Pura ed Applicata and Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Dipartimento di Matematica e FisicaUniversità degli Studi Roma TreRomeItaly

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