A. Babin and A. Figotin

Department of Mathematics

University of California at Irvine

Irvine, CA

USA

We develop a consistent mathematical
theory that describes nonlinear interaction of
wavepackets in a nonlinear periodic dielectric media for
the dimensions one, two and three. The theory is based
on the Maxwell equations with quadratic and cubic
constitutive relations. Solutions of the Maxwell
equations on long time intervals are expanded in
convergent series with respect to a small parameter
alpha, which measures the contribution of the
nonlinearity. After that we investigate in detail the
principal term of the expansion in the case where the
excitations (and, consequtively, solutions) have a form
of wavepackets. The ratio of the amplitude frequency and
the carrier frequency of a wavepacket is an important
small parameter rho. The principal term describes the
nonlinear interaction of a continuum of modes and is
written in the form of an oscillatory integral. The
phase function of the integral is written in terms of
the Floquet-Bloch dispersion relations of the periodic
media. We consider the situation when the stationary
phase method is applicable, that means that initially
the spatial extension of the wavepacket is larger then
the period and much smaller then the domain where the
medium is periodic. A detailed mathematical analysis
shows that the interaction integral expands into sum of
terms with different powers of rho and the leading terms
correspond to only a few interacting modes. We give a
classification of the nonlinear interactions between
wavepackets in a media with generic dispersion relations
based on the powers. The powers take on only a
relatively small number of prescribed values collected
in a table, their values depend on a type of degeneracy
of phase functions formed by the dispersion relations of
the media. The crucial role in selecting the strongest
interactions, in particular the second harmonic
generation in a quadratic medium, is played by internal
symmetries of the phase function.