  We study nonlinear systems of hyperbolic (in a wider sense) PDE's in entire
d-dimensional space describing wave propagation with the initial data in the
form of a finite sum of wavepackets referred to as multi-wavepackets. The
problem involves two small parameters beta and rho where: (i) (1/beta) is a
factor describing spatial extension of the wavepackets; (ii) (1/rho) is a
factor describing the relative magnitude of the linear part of the evolution
equation compared to its nonlinearity. For a wide range of the small parameters
and on time intervals long enough for strong nonlinear effects we prove that
multi-wavepackets are preserved under the nonlinear evolution. In particular,
the corresponding wave vectors and the band numbers of involved wavepackets are
"conserved quantities". We also prove that the evolution of a multi-wavepacket
is described with high accuracy by a properly constructed system of envelope
equations with a universal nonlinearity which in simpler cases turn into
well-known Nonlinear Schrodinger or coupled modes equations. The universal
nonlinearity is obtained by a certain time averaging applied to the original
nonlinearity. This can be viewed as an extension of the well known averaging
method developed for finite-dimensional nonlinear oscillatory systems to the
case of a general translation invariant PDE systems with the linear part having
continuous spectrum.
