Dynamics and Condensation of Complex Singularities For Burgers' Equation I

Abstract

Spatial analyticity properties of the solution to Burgers' equation with 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 viscosity 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 solution is a three-sheeted Riemann surface with three branch-point singularities. Exact pole locations are found independent of the viscosity at the inviscid shock time t* . For t \ne 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" algorithm to deal with the computationally intensive nonlinear interaction of the poles. Additionally, for t \leq t$, the small viscosity behavior of the poles is shown to be a perturbation of the conjugate inviscid branch-point singularities $\pm x_s(t)$. The numerical pole dynamics also provide the width of the analyticity strip which remains uniformly bounded away from zero, agreeing with asymptotic predictions. For small $\nu>0$ and t \geq t* , different saddle-point approximations of the solution are found within and outside the caustics $x=\pm x_s(t)$. The transition between the two regimes at $x=\pm x_s(t)$ is described by a uniform asymptotic expansion involving the Pearcey integral. The solution is computed for small viscosity using pole dynamics, finite differences (method of lines), and asymptotic methods (saddle-point method); numerical agreement is established.