Asymptotic integrability of nonlinear wave equations and the semiclassical limit of Lax pairs

We introduce the concept of asymptotic integrability of nonlinear wave equations, which means the integrability of Hamilton equations describing the propagation of a high-frequency wave packet along a smooth profile whose dynamics obeys the dispersionless limit of the original equations. We show that this limit case of complete integrability allows expressing the semiclassical limit of Lax pairs in terms of the dispersion law for linear waves and an integral of the Hamilton equations for the packet. If the Lax pair does not depend on derivatives of the wave variables, then the semiclassical limit coincides with the exact expressions. We illustrate the theory with specific examples.