Rayleigh–Taylor Instability of Fronts Described by the Nikolaevskii Equation

Rayleigh–Taylor instabilities take place as fluids of different densities are separated by a horizontal interface. The interface can be a reaction–diffusion front or an infinitesimal thin surface that can enhance or diminish the effects of buoyancy. Surface tension between immiscible fluids or autocatalytic reaction–diffusion fronts provides additional stability to the system. In other cases, such as fronts described by the Nikolaevskii equation, an additional mechanism for front instability exists. In confined domains, these fronts can be flat, oscillatory, or exhibit spatiotemporal chaos depending on domain widths. In the present work, we study fronts governed by the Nikolaevskii equation separating fluids of different density. We carry out a linear stability analysis of flat fronts using small perturbations of fixed wavelength. The dispersion relation between the wavenumber and the growth rate indicates transitions to instability for long wavelengths (type-II), a fixed wavelength (type-I), or both. We solve numerically the nonlinear equations, obtaining steady fronts constructed from solutions in smaller domains, or cells. We analyze the stability of these solutions in the presence of convection. We obtain the power spectrum for chaotic states for fronts propagating in larger domains.