Asymptotic integrability of nonlinear wave equations

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 to the quasiclassical limit of the Lax pair for the completely integrable equation having the corresponding dispersionless and linear dispersive behavior. We illustrate the theory with several examples.