DYNAMICS AND CONDENSATION OF COMPLEX SINGULARITIES FOR BURGERS-EQUATION .2.

The zero-viscosity limit of a meromorphic solution to Burgers' equatio n (BE) is found via an integral representation of the Mittag-Leffler e xpansion of the solution involving a ''polar'' measure. The weak zero- viscosity limit of this Borel measure (analogously to the zero-dispers ion Limit of the spectral measure in the Korteweg-de Vries (KdV) probl em) corresponds to the asymptotic density of poles which characterizes their condensation on the imaginary axis. The resulting integral repr esentation of the inviscid solution is computed by residues and is sho wn to match the characteristic solution up to the inviscid shock time t. The continuum limit of the Mittag-Leffler expansion and the Caloge ro dynamical system (CDS) (which describes the time evolution of the p oles) is a system of two integro-differential equations which provide a new representation of the solution to the inviscid BE. For t less th an or equal to t, a uniform asymptotic expansion of the Fourier trans form of the inviscid solution is obtained, thereby providing the analy ticity properties of the inviscid solution.