Seminar talk, 7 October 2015

From Geometry of Differential Equations
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Speaker: Alexey Samokhin

Title: On Burgers equation with a periodic boundary conditions on an interval

Asymptotics of the Burgers equation solutions on a finite interval with a periodic perturbation on the boundary are studied and a number of numeric illustrations is presented.

The equation describes a dissipative medium, so the initial constant profile passes into a travelling wave with a decreasing amplitude. In the case of a small viscosity the asymptotic profile looks like a saw (with periodic breaks of the derivative), similar to a known Fay solution on the half-line. Yet on an interval some new properties occur. The form of the solution retains the sawtooth profile yet its average over the interval differs from that on the half-line and depends also on the perturbation amplitude. Interaction between two perturbations of different frequencies is discussed, in particular the doubling of the envelope frequency. The figures were generated numerically using Maple PDETools package. It is worth noticing that standard procedures may easily loose stability at the points of derivative's discontinuity (e.g., at the tooth endpoint) leading to multi-oscillations and loss of precision. This problem was dealt with by adapting parameters of numeric simulations.