Global Solvability of the Anharmonic Oscillator Model from Nonlinear Optics

Abstract

The field equations describing the propagation of electromagnetic waves in a nonlinear dielectric medium whose polarization responds locally to the electric field as an anharmonic oscillator with potential $V(P)$ have smooth solutions global in space and time for arbitrary smooth initial data as soon as V has bounded derivatives of order less than or equal to three. This is true in spite of the fact that solutions of the nonlinear Shrödinger equation which approximate the fields in the slowly varying envelope approximation may blow up in finite time.