Dynamics and Condensation of Complex Singularities for Burgers' Equation II

Abstract

The zero-viscosity limit of a meromorphic solution to Burgers' equation (BE) is found via an integral representation of the Mittag--Leffler expansion of the solution involving a "polar" measure. The weak zero-viscosity limit of this Borel measure (analogously to the zero-dispersion limit of the spectral measure in the Korteweg--de Vries (KdV) problem) corresponds to the asymptotic density of poles which characterizes their condensation on the imaginary axis. The resulting integral representation of the inviscid solution is computed by residues and is shown to match the characteristic solution up to the inviscid shock time t*. The continuum limit of the Mittag--Leffler expansion and the Calogero dynamical system (CDS) (which describes the time evolution of the poles) is a system of two integro-differential equations which provide a new representation of the solution to the inviscid BE. For t \leq t*, a uniform asymptotic expansion of the Fourier transform of the inviscid solution is obtained, thereby providing the analyticity properties of the inviscid solution.