Научный вестник МГТУ ГА (Nov 2016)
SAWTOOTH SOLUTIONS TO THE BURGERS EQUATION ON AN INTERVAL
Abstract
The asymptotic behavior of solutions of the Burgers equation and its generalizations with initial value - boundary problem on a finite interval with constant boundary conditions is studied. Since the equation describes the movement in a dissipative medium, the initial profile of the solution will evolve to an time-invariant solution with the same boundary values. However there are three ways of obtaining the same result: the initial profile may regularly decay to the smooth invariant solution; or a Heaviside-type gap develops through a dispersive shock and multi-oscillations; or an asymptotic limit is a stationary ’sawtooth’ solution with periodical breaks of derivative.