The Rayleigh-Taylor Instability and Nonlinear Waves

A nonlinear stage of the two-dimensional Rayleigh-Taylor instability is studied by including the effect of surface tension between the two fluids. By means of the reductive perturbation method, three types of nonlinear evolution equations for the interface are obtained. Each equation is valid within a certain region of the wave number k introduced in the linearized theory. When k is sufficiently large and thus in the stable region, the nonlinear Schrodinger equation is derived. Depending on the value of k in this region, bright or dark solitons exist. When k is nearly equal to the critical wave number k c , the unstable nonlinear Schrodinger equation is obtained. Further, it is shown that a nonlinear diffusion equation is obtained in the unstable region where k is smaller than k c .