Stationary solutions of the one-dimensional nonlinear Schroedinger equation: I. Case of repulsive nonlinearity
Abstract
In this first of two papers, we elucidate properties of all stationary solutions of the nonlinear Schroedinger equation with constant external potential on a finite one dimensional interval, with both box and periodic boundary conditions, for the case of repulsive nonlinearity. This case also describes standing waves in optical fibers in the defocusing regime. The companion paper provides the same treatment for the case of attractive nonlinearity. Such solutions can all be expressed in terms of Jacobian elliptic functions. Our solutions take the form of stationary trains of dark or grey solitons. These solutions give insight into the the types of solitonic dynamics which may arise in collisions between independently-prepared condensates. Under box boundary conditions, these solutions are the bounded analog of dark solitons on the infinite line, and are in one-to-one correspondence with the usual particle-in-a-box solutions of the linear Schroedinger equation. Under periodic boundary conditions, we find several classes of solutions: the nonlinear version of the well-known, real particle-on-a-ring solutions in linear quantum mechanics; constant amplitude, plane wave solutions corresponding to boosts of the condensate, which are the nonlinear version of the complex particle-on-a-ring solutions; and a novel class of intrinsically complex, nodeless solutions which are the bounded analog of grey solitons on the infinite line. As density notches may be placed anywhere on the ring, they provide a class of symmetry-breaking solutions which have a high degeneracy, as is the case for symmetry-breaking vortex solutions in two dimensions.Keywords
All Related Versions
This publication has 0 references indexed in Scilit: