NONLINEAR SELF-MODULATION IN NEWTONIAN GRAVITY

We study quasi-nonlinear evolution of the density perturbation in Newt onian gravity. Weak mode-mode coupling in a small range below the Jean s wavelength is considered. In order to extract nonlinear dynamics we utilize a reductive perturbation, which is well known in mechanics and hydrodynamics and improves a naive perturbation. We show that the bas ic equations for the acoustic wave reduce to a nonlinear Schrodinger e quation. It describes a competition between dispersion originated from gravitational attraction and nonlinearity up to cubic order of the am plitude of the acoustic wave. In purely 1-dimensional motion, there ex ists localized structures as soliton solutions of two distinctive type s depending on the wavelength. More interesting is an instability pres ent in 3-dimensional motion. Namely, a progressive wave is unstable un der a long-wave perturbation transverse to the direction of progressio n. It may imply a possible nonlinear growth of the density fluctuation below the Jeans scale.