On the Construction of a Modulating Multiphase Wavetrain for a Perturbed Kdv Equation

This paper summarizes the status of a direct construction of an asymptotic representation of a modulating multiphase wavetrain for a class of perturbed kdV equations. This class includes the kdV-Burgers equation. The calculations apply on a “boundary” between dispersive and dissipative behavior. The construction proceeds by standard asymptotic methods. The result of the construction is an invariant representation of the reduced equations which permits their diagonalization. While mathematically the construction is incomplete, care is taken to identify the mathematical status of each step in the construction. The equivalence of this constructive approach with postulated averages of conservation laws is established for two phase waves. Finally, the Young measure for this program is constructed explicitly.