Semiclassical Asymptotics for Weakly Nonlinear Bloch Waves

Abstract

Abstract. We study the simultaneous semi-classical and adiabatic asymptotics for a class of (weakly) nonlinear Schrödinger equations with a fast periodic potential and a slowly varying confinement potential. A rigorous two-scale WKB-analysis, locally in time, is performed. The main nonlinear phenomenon is a modification of the Berry phase.

Multiphase semiclassical approximation of the one-dimensional harmonic crystal: I. The periodic case

“…We assume the reader to be familiar with Bloch's theory; a comprehensive presentation is given in [1], see also [7,8,14,15,10]. In this section, we shall show that for simple enough displacements (2) (typically with few vibration modes), it is possible to derive a WKB approximation for the wave function ψ solution of (3) for smooth V per .…”

“…Plugging the approximation A(t, x, x/ε) exp(iϕ(t, x)/ε) inside (4) and balancing the O(1) terms following carefully [7,8,14] can be shown to lead to an Hamilton-Jacobi equation for the phase:…”

“…The principal amplitude stems from balancing O(ε) terms and is usually handled by means of the "Feschbach method", see the paper [7] to which we refer for a detailed presentation of the computation. Here we shall limit ourselves to specify the changes occuring in the derivation because of the time-dependence ofṼ per .…”

“…Here we shall limit ourselves to specify the changes occuring in the derivation because of the time-dependence ofṼ per . Actually, everything proceeds as in the Appendix A of [7], except for the last step that we explain now §: let us denote L the usual geometric optics transport operator associated to the Bloch theory:…”

“…The principal amplitude is defined as a 0 (t, x) such that there holds: [7], we obtain the following equation for a 0 :…”