On nonlinear dynamics of neutral modes in elastic waves in granular media

We analyse a passive system featuring a neutrally stable short-wavelength mode. The system is modelled by the Nikolaevskiy equation relevant to certain type of elastic waves, reaction-diffusion systems and convection. Due to the nonlinear coupling between the time-dependent Fourier modes, the system exhibits asymptotically slow evolution towards either zero or non-zero steady state depending on the initial condition and the neutral wave number. Using the centre manifold technique, we deduced that the decay law is that of inverse square root of time. The result is confirmed by the computations of the dynamical system for the Fourier modes.