Starting from the three-dimensional Gross-Pitaevskii equation (3D GPE) we derive a 1D generalized nonpolynomial Schrödinger equation (1D g-NPSE) which describes the dynamics of Bose-Einstein condensates under the action of a generic potential in the longitudinal axial direction and of an anisotropic harmonic potential in the transverse radial direction. This equation reduces to the familiar 1D ...

An adaptive numerical method for solution of the time-dependent Schrödinger equation is presented. By using an interpolating wavelet transform the number of used points can be reduced, constructing an adaptive sparse point representation. Two di erent discretisation approaches are studied in 1D ; one point-based and one based on equidistant blocks. Due to the nature of the wavelet transform a s...

In one-dimensional quantum mechanics, or the Sturm-Liouville theory, Crum’s theorem describes the relationship between the original and the associated Hamiltonian systems, which are iso-spectral except for the lowest energy state. Its counterpart in ‘discrete’ quantum mechanics is formulated algebraically, elucidating the basic structure of the discrete quantum mechanics, whose Schrödinger equa...

In this paper we establish the orbital stability of standing wave solutions associated to one-dimensional Schrödinger-Kirchhoff equation. The presence a mixed term gives us more dispersion, and consequently, different scenario for solitary waves in contrast with corresponding nonlinear Schrödinger For periodic waves, exhibit two explicit prove energy space.

A selfconsistent model for charged particles, accounting for quantum confinement, diffusive transport and electrostatic interaction is considered. The electrostatic potential is a solution of a three dimensional Poisson equation with the particle density as the source term. This density is the product of a two dimensional surface density and that of a one dimensional mixed quantum state. The su...

We report an algorithm of constructing linear and nonlinear potentials in the two-dimensional Gross-Pitaevskii equation subject to given boundary conditions, which allow for exact analytic solutions. The obtained solutions represent superfluid flows in inhomogeneous Bose-Einstein condensates. The method is based on the combination of the similarity reduction of the two-dimensional Gross-Pitaevs...

A generalized inverse scattering method has been developed for arbitrary n dimensional Lax equations. Subsequently, the method has been used to obtain N soliton solutions of a vector higher order nonlinear Schrödinger equation, proposed by us. It has been shown that under suitable reduction, vector higher order nonlinear Schrödinger equation reduces to higher order nonlinear Schrödinger equatio...

Using a standing light wave potential, a stable quasi-one-dimensional attractive dilute-gas Bose-Einstein condensate can be realized. In a mean-field approximation, this phenomenon is modeled by the cubic nonlinear Schrödinger equation with attractive nonlinearity and an elliptic function potential of which a standing light wave is a special case. New families of stationary solutions are presen...

Using conformal coordinates associated with projective conformal relativity we obtain a conformal Klein-Gordon partial differential equation. As a particular case we present and discuss a conformal ‘radial’ d’Alembert-like equation. As a by-product we show that this ‘radial’ equation can be identified with a one-dimensional Schrödinger-like equation in which the potential is exactly the second ...

We systematically provide a similarity transformation reducing the (3 + 1)-dimensional inhomogeneous coupled nonlinear Schrödinger (CNLS) equation with variable coefficients and parabolic potential to the (1 + 1)-dimensional coupled nonlinear Schrödinger equation with constant coefficients. Based on the similarity transformation, we discuss the dynamics of the propagation of the three-dimension...