Pole condensation and the Riemann surface associated with a shock in Burgers' equation

Abstract

Using a paradigmatic initial condition, we completely analyse the spatial analytic structure of Burgers velocity field and its dependence on time t and viscosity v. The viscous solution has an infinite number of complex poles appearing suddenly at t = 0+. When v ↓ 0 these poles condense on Stokes lines giving rise to the inviscid singularities ; the 3-sheeted Riemann surface associated with the latter provides an analytic path across the jump of the shock.