An example of chaos for a cubic nonlinear Schrödinger equation with periodic inhomogeneous nonlinearity

Using a topological approach we prove the existence of infinitely many periodic solutions, as well as the presence of symbolic dynamics for second order nonlinear differential equations of the form −¨u − μu + g(t)h(u) = 0 where μ > 0 is a given constant and g : R → R is a periodic positive weight function. Our main application concerns the study of the case in which h(u) is a cubic nonlinearity. Such a choice is motivated by previous investigations dealing with the nonlinear Schr¨odinger equation iψt = −1/2ψxx + g(x)|ψ|2ψ.