  Due to the nonlinearity of the Euler{Poisson equations, it is possible that
the nonlinear Jeans instability may lead to a faster density growing rate than
the rate in the standard theory of linearized Jeans instability, which
motivates us to study the nonlinear Jeans instability. The aim of this article
is to develop a method proving the Jeans instability for slightly nonlinear
Euler-Poisson equations in the expanding Newtonian universe. The standard
proofs of the Jeans instability rely on the Fourier analysis. However, it is
difficult to generalize Fourier method to a nonlinear setting, and thus there
is no result in the nonlinear analysis of Jeans instability. We firstly develop
a non-Fourier-based method to reprove the linearized Jeans instability in the
expanding Newtonian universe. Secondly, we generalize this idea to a slightly
nonlinear case. This method relies on the Cauchy problem of the Fuchsian system
due to the recent developments of this system in mathematics. The fully
nonlinear Jeans instability for the Euler-Poisson and Einstein-Euler equations
are in progress.
