Quasiperiodic Solutions in Weakly Nonlinear Gas Dynamics. Part I. Numerical Results in the Inviscid Case

We exhibit and study a new class of solutions for the one-dimensional inviscid Euler equations of Gas Dynamics in a bounded domain with reflecting boundary conditions, in the weakly nonlinear regime. These solutions do not present the usual wave breaking leading to shock formation, even though they have nontrivial acoustic components and operate in the nonlinear regime. We also show that these 'Non Breaking for All Times' (NBAT) solutions are globally attracting for the long time evolution of the equations.
The Euler equations of Gas Dynamics (in the weakly nonlinear regime with reflecting boundary conditions) can be reduced to an inviscid Burgers-like equation for the acoustic component, with a linear integral self-coupling term and periodic boundary conditions. The integral term arises as a result of the nonlinear resonant interactions of the sound waves with the entropy variations in the flow. This integral term turns out to be weakly dispersive. The NBAT solutions arise as a result of the interplay of this dispersion with the 'standard' wave-breaking nonlinearity in the Burgers equation.
In addition to the previously known weakly nonlinear standing acoustic wave NBAT solutions, we found a family of new, never-breaking, attracting solutions by direct numerical simulation. These are quasiperiodic in time with two periods. In phase space these solutions lie on a surface 'centered' around the standing waves. Only two standing-wave solutions (the maximum amplitude and the trivial vanishing wave) are in the attracting set. All of the others are quasiperiodic in time with two periods.