Quantum nonlinear Schrödinger equation: Two solutions

David J. Fernández C., O. Pavón-Torres

By employing supersymmetric quantum mechanics, we present a general algorithm to construct supersymmetric partner potentials and hence derive exact stationary solutions of the inhomogeneous nonlinear Schrödinger equation (INLSE). This is possible due to the connection between the INLSE and the nonlinear Schrödinger equation (NLSE), which can be established from a treatment based on Lie point symmetries and is related with Schrödinger equation, under certain conditions. As an illustrative example, we construct exact solutions for the INLSE through a Pösch-Teller potential with a single bound state.

N. M. Bogoliubov, V. E. Korepin

Jochen Pade

Pang Xiao‐feng

The dynamic and collision features of microscopic particles described by nonlinear Schrodinger equation are investigated deeply using the analytic and the Runge-Kutta method of numerical simulation. The results show that the microscopic particles have a wave-corpuscle duality and are stable in propagation. When the two microscopic particles are collided, they can go through each other and retain their form after their collision of head-on from opposite directions, This feature is the same with that of the classical particles. However, a wave peak of large amplitude, which is a result of complicated superposition of two solitary waves, occurs in the colliding process. This displays the wave feature of microscopic particles. Therefore, the collision process shows clearly that the solutions of the nonlinear Schrodinger equation have a both corpuscle and wave feature, then the microscopic particles represented by the solutions have a wave-corpuscle duality. Obviously, this is due to the nonlinear interaction b||2. Thus we can determine the nonlinear Schrodinger equation can describe correctly the natures and properties of microscopic particles in quantum systems. Key words: Microscopic particle Schrodinger equation; Wave-corpuscle duality; Nonlinear interaction; Collision; Propagation; Quantum mechanics

David W. Brown, Katja Lindenberg, Bruce J. West

Bernard Deconinck, Béla A. Frigyik, J. Nathan Kutz

The cubic nonlinear Schrodinger equation with repulsive nonlinearity and elliptic function potential in two-dimensions models a repulsive dilute gas Bose--Einstein condensate in a lattice potential. A family of exact stationary solutions is presented and its stability is examined using analytical and numerical methods. All stable trivial-phase solutions are off-set from the zero level. Our results imply that a large number of condensed atoms is sufficient to form a stable, periodic condensate. Key words: Bose-Einstein condensates, two-dimensional nonlinear Schrodinger equation, periodic potential, elliptic functions PACS: 65.65 1 Introduction The cubic nonlinear Schrodinger equation with attractive or repulsive nonlinearity and a potential is used as a mean-field model for the dynamics of a dilute-gas Bose Einstein condensate (BEC) [1,2]. In this case, the equation is often referred to as the Gross-Pitaevskii equation. These BECs are of interest in both the theoretical and experim...