"Large amplitude nonlinear resonant acoustic waves without shocks."

Rodolfo Ruben Rosales (MIT)

We consider the problem of describing the "long" time asymptotic behavior for solutions of the one dimensional Euler equations of Gas Dynamics in a bounded domain (say, a tube with rigid walls at the ends, equivalent to a periodic initial conditions situation). Time is measured in terms of an acoustic transit period and "long" typically turns out to mean a few hundred such periods.

Intuitively one would expect that any acoustic component would lead to shocks that will then induce rapid decay into it. Thus, the long time behavior should be described by a "pure entropy wave". This appears to be false. We have constructed examples of solutions with large nontrivial acoustic component and no shocks at all ( these examples are, in fact, periodic in time, thus they have no decay ). Numerical checks of stability for these examples indicate that in fact they are stable. Furthermore, starting from "arbitrary" initial conditions, careful numerical calculations show that the solutions (after an initial "transient" period with shocks) eventually settle down to generally quasiperiodic (in time) solutions with nontrivial acoustical component.

These results and other related ones will be presented. We are also investigating the effects of a small amount of dissipation in the equations. In this case, basically what seems to happen is that, after the "transient" period with shocks, the solution settles down to a quasiperiodic (in time) solution whose parameters slowly drift as the energy is dissipated.