Complex dynamics in one-dimensional nonlinear Schrödinger equations with stepwise potential (Q1723093)

Summary: We prove the existence and multiplicity of periodic solutions as well as solutions presenting a complex behavior for the one-dimensional nonlinear Schrödinger equation \(-\varepsilon^2 u''+V(x)u=f(u),\) where the potential \(V(x)\) approximates a two-step function. The term \(f(u)\) generalizes the typical \(p\)-power nonlinearity considered by several authors in this context. Our approach is based on some recent developments of the theory of topological horseshoes, in connection with a linked twist maps geometry, which are applied to the discrete dynamics of the Poincaré map. We discuss the periodic and the Neumann boundary conditions. The value of the term \(\varepsilon >0\), although small, can be explicitly estimated.