Complex Dynamics in One-Dimensional Nonlinear Schrödinger Equations with Stepwise Potential

Abstract EN

We prove the existence and multiplicity of periodic solutions as well as solutions presenting a complex behavior for the one-dimensional nonlinear Schrödinger equation -ε2u′′+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 ε>0, although small, can be explicitly estimated.