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

Abstract

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.

Multiple homoclinic solutions for a one-dimensional Schrödinger equation

“…The equation 1 is the very classical one-dimensional Schrödinger equation, which comes from the study of stationary waves of a nonlinear Schrödinger equation in R N ; there is a huge literature on this equation and several different results have been proved (see for instance, among the others, [1], [5], [8], [20]). Very recently, some results on the lines of our approach have been proved in [24]; indeed, in the quoted paper the authors study the existence of chaotic dynamics for the equation 1, using the S.A.P. method.…”

“…where f (x) = xh(x), for every x ∈ R, and h ∈ C 1 (R; R) is such that h(0) = 0, h (x) > 0 for every x > 0 and lim x→+∞ h(x) = +∞ (cfr. [24]). The proof of Theorem 1.1 is based on the combination of the Conley-Ważewski's method (see [4,23]) and of the so-called S.A.P.…”