  We introduce the notion of asymptotic integrability into the theory of
nonlinear wave equations. It means that the Hamiltonian structure of equations
describing propagation of high-frequency wave packets is preserved by
hydrodynamic evolution of the large-scale background wave, so that these
equations have an additional integral of motion. This condition is expressed
mathematically as a system of equations for the carrier wave number as a
function of the background variables. We show that a solution of this system
for a given dispersion relation of linear waves is related with the
quasiclassical limit of the Lax pair for the completely integrable equation
having the corresponding dispersionless and linear dispersive behavior. We
illustrate the theory by several examples.
