# "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.