Conventionally, bright solitary wave solutions can be obtained in self-focusing nonlinear Schrödinger equations with attractive self-interaction. However, when selfinteraction becomes repulsive, it seems impossible to have bright solitary wave solution. Here we show that there exists symbiotic bright solitary wave solution of coupled nonlinear Schrödinger equations with repulsive self-interacti...

Starting from the forward and backward infinitesimal generators of bilateral, timehomogeneous Markov processes, the self-adjoint Hamiltonians of the generalized Schrödinger equations are first introduced by means of suitable Doob transformations. Then, by broadening with the aid of the Dirichlet forms, the results of the Nelson stochastic mechanics, we prove that it is possible to associate bil...

The possibility of finite-time, dispersive blow up for nonlinear equations of Schrödinger type is revisited. This mathematical phenomena is one of the possible explanations for oceanic and optical rogue waves. In dimension one, the possibility of dispersive blow up for nonlinear Schrödinger equations already appears in [9]. In the present work, the existing results are extended in several ways....

We prove a representation formula for solutions of Schrödinger equations with potentials multiplied by a temporal real-valued white noise in the Stratonovich sense. Using this formula, we obtain a dispersive estimate which allows us to study the Cauchy problem in L or in the energy space of model equations arising in Bose Einstein condensation [1] or in fiber optics [2]. Our results also give a...

We consider the relation between the discrete coupled nonlinear Schrödinger equation and Toda equation. Introducing complex times we can show the intergability of the discrete coupled nonlinear Schrödinger equation. In the same way we can show the integrability in coupled case of dark and bright equations. Using this method we obtain several integrable equations.

We give a new proof of Hardy’s uncertainty principle, up to the end-point case, which is only based on calculus. The method allows us to extend Hardy’s uncertainty principle to Schrödinger equations with non-constant coefficients. We also deduce optimal Gaussian decay bounds for solutions to these Schrödinger equations.

We present a new discrete Adomian decomposition method to approximate the theoretical solution of discrete nonlinear Schrödinger equations. The method is examined for plane waves and for single soliton waves in case of continuous, semi– discrete and fully discrete Schrödinger equations. Several illustrative examples and Mathematica program codes are presented.

We give a new proof of a theorem of Bourgain [4], asserting that solutions of linear Schrödinger equations on the torus, with smooth time dependent potential, have Sobolev norms growing at most like t when t→ +∞, for any > 0. Our proof extends to Schrödinger equations on other examples of compact riemannian manifolds.

The spectrum of a Schrödinger operator with a perfect honeycomb lattice potential has special points, called Dirac points where the lowest two branches of the spectrum touch. Deformations can result in the merging and disappearance of the Dirac points and the originally intersecting dispersion relation branches separate. Corresponding to these deformations, nonlinear envelope equations are deri...