SAWTOOTH SOLUTIONS TO THE BURGERS EQUATION ON AN INTERVAL

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.