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Stationary Solutions of the One-Dimensional Nonlinear Schrodinger Equation: I. Case of Repulsive Nonlinearity



L D. Carr, Charles W. Clark, W P. Reinhardt


In this first of two papers, we elucidate properties of all stationary solutions of the nonlinear Schr dinger equation with constant external potential on a finite one dimensional interval, with both box and periodic boundary conditions, for the case of repulsive nonlinearity. 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. The case of repulsive nonlinearity describes a dilute Bose-Einstein condensate with repulsive pair atomic interactions. The solutions given here provide an exact mean-field description of collective excitations of such condensation one dimension, and approximate longitudinal excitations in three-dimensional traps of high aspect ratio. This case also describes standing waves in optical fibers in the defocusing regime. Our solutions take the form of stationary trains of dark or grey solitons. These solutions give insight into 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 Schr dinger 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 solitions 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.
Physical Review A (Atomic, Molecular and Optical Physics)
No. 6


Bose-Einstein condensation, Jacobian elliptic function, nonlinear Schr dinger equation, solitary wave, soliton


Carr, L. , Clark, C. and Reinhardt, W. (2000), Stationary Solutions of the One-Dimensional Nonlinear Schrodinger Equation: I. Case of Repulsive Nonlinearity, Physical Review A (Atomic, Molecular and Optical Physics) (Accessed May 29, 2024)


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Created November 30, 2000, Updated October 12, 2021