DYNAMICS AND CONDENSATION OF COMPLEX SINGULARITIES FOR BURGERS-EQUATION .1.

Spatial analyticity properties of the solution to Burgers' equation wi th a generic initial data are presented, following the work of Bessis and Fournier [Research Reports in Physics: Nonlinear Physics, Springer -Verlag, Berlin, Heidelberg, 1990, pp. 252-257]. The positive viscosit y solution is a meromorphic function with a countable set of conjugate poles confined to the imaginary axis. Their motion is governed by an infinite-dimensional Calogero dynamical system (CDS). The inviscid sol ution is a three-sheeted Riemann surface with three branch-point singu larities. Exact pole locations are found independent of the viscosity at the inviscid shock time t. For t not equal t*, the time evolution of the poles is obtained numerically by solving a truncated version of the CDS. A Runge-Kutta scheme is used together with a ''multipole'' a lgorithm to deal with the computationally intensive nonlinear interact ion of the poles. Additionally, for t less than or equal to t, the sm all viscosity behavior of the poles is shown to be a perturbation of t he conjugate inviscid branch-point singularities +/-x(s)(t). The numer ical pole dynamics also provide the width of the analyticity strip whi ch remains uniformly bounded away from zero, agreeing with asymptotic predictions.For small v > 0 and t greater than or equal to t, differe nt saddle-point approximations of the solution are found within and ou tside the caustics x = +/-x(s)(t). The transition between the two regi mes at x = +/-x(s)(t) is described by a uniform asymptotic expansion i nvolving the Pearcey integral. The solution is computed for small visc osity using pole dynamics, finite differences (method of lines), and a symptotic methods (saddle-point method); numerical agreement is establ ished.