Spectral averaging of small-amplitude sine-Gordon wave trains.

It is well known that the nonlinear Schroedinger equation is the generic envelope description of nonlinear wave trains in the small-amplitude limit. V. E. Zakharov and E. A. Kuznetsov (Physica 18D, 455 (1986)) have shown that for many systems integrable via inverse-scattering techniques it is possible, through the use of multiscale techniques, to derive the nonlinear Schroedinger Lax pair from the Lax pair of the system that is being modulated. It will be shown that this technique of ''multiscale averaging'' can be applied to the sine-Gordon theory to obtain not only the nonlinear Schroedinger Lax pair from the sine-Gordon Lax pair, but also the nonlinear Schroedinger spectral data, conservation laws, THETA-function solutions, and reality constraint from their sine-Gordon counterparts. This allows a physical interpretation of the mathematical elements of the nonlinear Schroedinger envelope in terms of the physical characteristics of the sine-Gordon system that is being modulated.