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Stationary Solutions of the One-Dimensional Nonlinear Schrodinger Equation: II. Case of Attractive Nonlinearity



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


In this second of two papers, we present all stationary solutions of the nonlinear Schrodinger equation with box of periodic boundary conditions for the case of attractive nonlinearity. Our solutions take the form of stationary trains of bright silitons. Under box boundary conditions the solutions are the bounded analog of bright solitons on the infinite line, and are in one-to-one correspondence with particle-in-a-box solutions to the linear Schrodinger equation. Under periodic boundary conditions we find several classes of solutions: constant amplitude solutions corresponding to boosts of the condensate; the nonlinear version of the well-known particle-on-a-ring solutions in linear quantum mechanics; nodeless, real solutions; and a novel class of intrinsically complex, nodeless solutions. The set of such solutions on the ring is described by the Cn character tables from the theory of point groups. We make experimental predictions about the form of the ground state and modulational instability. We show that, though this is the analog of some of the simplest problems in linear quantum mechanics, nonlinearity introduces new and surprising phenomena in the stationary, one-dimensional, non-linear Schrodinger equation. We also note that in various limits the spectrum of the nonlinear Schrodinger equation reduces to that of the box, the Rydberg, and the harmonic oscillator, the latter being for repulsive nonlinearity, thus including the three most common and important cases of linear quantum mechanics.
Physical Review A (Atomic, Molecular and Optical Physics)
No. 6


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


Carr, L. , Clark, C. and Reinhardt, W. (2000), Stationary Solutions of the One-Dimensional Nonlinear Schrodinger Equation: II. Case of Attractive Nonlinearity, Physical Review A (Atomic, Molecular and Optical Physics) (Accessed November 30, 2023)
Created November 30, 2000, Updated October 12, 2021