This paper deals with one-dimensional (1D) nonlinear Schrödinger equation with a multiplicative potential, subject to Dirichlet boundary conditions. It is proved that for each prescribed integer b > 1, the equation admits smallamplitude quasi-periodic solutions, whose b-dimensional frequencies are small dilation of a given Diophantine vector. The proof is based on a modified infinitedimensional...

We consider in this work the initial value problem for the one dimensional cubic nonlinear Schrödinger equation. In order to integrate it numerically, one option frequently used, is to impose local absorbing boundary conditions. A finite element discretization in space of the cubic nonlinear Schrödinger equation is considered along with the absorbing boundary conditions obtained for an analogou...

In this paper, one-dimensional (1D) nonlinear Schrödinger equation iut − uxx + |u|2pu= 0, p ∈N, with periodic boundary conditions is considered. It is proved that the above equation admits small-amplitude quasi-periodic solutions corresponding to 2-dimensional invariant tori of an associated infinite-dimensional dynamical system. The proof is based on infinite-dimensional KAM theory, partial no...

We present a direct and rather elementary method for defining and analyzing one-dimensional Schrödinger operators H = −d2/dx2 + μ with measures as potentials. The basic idea is to let the (suitably interpreted) equation −f ′′+μf = zf take center stage. We show that the basic results from direct and inverse spectral theory then carry over to Schrödinger operators with measures.