Analytic approaches are developed for integrating the nondiagonalizable Whi
tham equations for the generation and propagation of nonlinear modulated fi
nite-amplitude waves in dissipationless dispersive media. Natural matching
conditions for these equations are stated in a general form analogous to th
e Gurevich-Pitaevskii conditions for the averaged Korteweg-de Vries equatio
ns. Exact relationships between the hydrodynamic quantities on different si
des of a dissipationless shock wave, an analog of the shock adiabat in ordi
nary dissipative hydrodynamics and first proposed on the basis of physical
considerations by Gurevich and Meshcherkin,(4) are obtained. The boundaries
of a self similar, dissipationless shock wave are determined analytically
as a function of the density jump. Some specific examples are considered. (
C) 1999 American Institute of Physics. [S1063-7761(99)02603-7].