## New solutions for nonlinear perfect fluids.

##### Abstract

We investigate the Einstein system that governs the evolution of uncharged shear-free
spherically symmetric fluids. First we present the Einstein equations for the static
spherically symmetric gravitational fields in isotropic coordinates. Also the nonstatic
spherically symmetric gravitational fields are studied. We have demonstrated that
the fundamental differential equation governing the behaviour of the model is of the
Emden-Fowler type. Such equations also arise in applications in Newtonian physics.
The field equations governing the gravitational behaviour of the model are generated.
We integrate the system of partial differential equations and apply a transformation
that reduces the system to a second order ordinary differential equation. To solve
the resulting ordinary differential equation we employ the method of characteristics to
find different expressions for the gravitational potentials. We employ the method of
characteristics to obtain first integrals for the Emden-Folwer type equation. To apply
the method, we make use of the associated multipliers which are obtained via the Euler
operator acting on the arbitrary multiplier and differential equation. These multipliers
can be obtained under the various forms of the arbitrary function representing the
gravitational potential under which the equation becomes integrable. Thus expanding
the differential equation with the associated multiplier, we can find first integrals by
solving the system of partial differential equations. The study is comprised of various
forms of the multipliers associated to first integrals of the equation in question.