Decay in Systems with Neutral Short-Wavelength Stability: The Presence of a Zero Mode - Ali, Adham A.; Ahmed, Fatima Z. (Discontinuity, Nonlinearity, and Complexity, 2026)
A characteristic feature of seismic waves is the presence of dominant frequency/wave number in the spectrum. A well-known model for such waves is the Nikolaevskiy equation, which is also applicable to some reaction-diffusion systems and Rayleigh-Benard convection. For the critical case when there is one neutral mode, we describe the dynamics of the Fourier modes (elastic waves) under the Nikolaevskiy equation using the centre manifold technique. After quickly attracted to the surface (manifold), the modes then evolve slow algebraic decay. An inverse square-root law for the decaying regime is obtained. The result is confirmed by direct computations of the dynamical system for the modes.