Dynamics of the nonlinear Klein-Gordon equation in the nonrelativistic limit, II

Abstract

We study the 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 an order-$r$ normalized approximation of NLKG (which corresponds to the NLS at order $r=1$), and prove that when $M={\mathbb{R}}^d$, $d\ge 2$, small radiation solutions of the order-$r$ normalized equation approximate solutions of the nonlinear NLKG up to times of order $\mathcal{O}(c^{2(r-1)})$.

more details added to the proof of Proposition 5.4 ; some typos corrected

References in corpus (3)

• Resonances, Radiation Damping and Instability in Hamiltonian Nonlinear Wave Equations
• Wellposedness of Cauchy problem for the Fourth Order Nonlinear Schrödinger Equations in Multi-dimensional Spaces
• Global estimates of fundamental solutions for higher-order Schrödinger equations