The Bose-Einstein condensate (BEC) of a dilute gas of bosons is well described by the three-dimensional Gross-Pitaevskii equation (3D GPE), that is a nonlinear Schrödinger equation. By imposing a transverse confinement the BEC can travel only in the cylindrical axial direction. We show that in this case the BEC with attractive interaction admits a 3D bright soliton solution which generalizes th...

The potentials for a one dimensional Schrödinger equation that are displaced along the x axis under second order Darboux transformations, called 2-SUSY invariant, are characterized in terms of a differential-difference equation. The solutions of the Schrödinger equation with such potentials are given analytically for any value of the energy. The method is illustrated by a two-soliton potential....

The aim of this paper is to develop new optimized Schwarz algorithms for the one dimensional Schrödinger equation with linear and nonlinear potential. classical algorithm an iterative process. In case time-independent potential, we construct explicitly interface problem use direct LU method on problem. therefore turns be a Thus, independent transmission condition numerical computation smaller. ...

A general method is proposed to build exact artificial boundary conditions for the one-dimensional nonlocal Schrödinger equation. To this end, we first consider spatial semi-discretization of equation, and then develop an accurate numerical computing Green's function semi-discrete These functions are next used corresponding model. Numerical results illustrate accuracy conditions. The methodolog...