  A nonlinear Schrodinger equation arising from light propagation down an
inhomogeneous medium is considered. The inhomogeneity is reflected through a
non-uniform coefficient of the non-linear term in the equation. In particular,
a combination of self-focusing and self-defocusing nonlinearity, with the
self-defocusing region localized in a finite interval, is investigated. Using
numerical computations, the extension of linear eigenmodes of the corresponding
linearized system into nonlinear states is established, particularly nonlinear
continuations of the fundamental state and the first excited state. The
(in)stability of the states is also numerically calculated, from which it is
obtained that symmetric nonlinear solutions become unstable beyond a critical
threshold norm. Instability of the symmetric states is then investigated
analytically through the application of a topological argument. Determination
of instability of positive symmetric states is reduced to simple geometric
properties of the composite phase plane orbit of the standing wave. Further the
topological argument is applied to higher excited states and instability is
again reduced to straightforward geometric calculations. For a relatively high
norm, it is observed that asymmetric states bifurcate from the symmetric ones.
The stability and instability of asymmetric states is also considered.
