Diffusive N-Waves and Metastability in the Burgers Equation

Abstract

We study the effect of viscosity on the large time behavior of the viscous Burgers equation by using a transformed version of Burgers (in self-similar variables) that captures efficiently the mechanism of transition to the asymptotic states and allows us to estimate the time of evolution from an N-wave to the final stage of a diffusion wave. Then we construct certain special solutions of diffusive N-waves with unequal masses. Finally, using a set of similarity variables and a variant of the Cole--Hopf transformation, we obtain an integrated Fokker--Planck equation. The latter is solvable and provides an explicit solution of the viscous Burgers equation in a series of Hermite polynomials. This format captures the long-time--small-viscosity interplay, as the diffusion wave and the diffusive N-waves correspond, respectively, to the first two terms in the Hermite polynomial expansion.