All stationary solutions to the one-dimensional nonlinear Schrödinger equation under box or periodic boundary conditions are presented in analytic form for the case of attractive nonlinearity. A companion paper treated the repulsive case. Our solutions take the form of bounded, quantized, stationary trains of bright solitons. Among them are two uniquely nonlinear classes of nodeless solutions, whose properties and physical meaning are discussed in detail. The full set of symmetry-breaking stationary states are described by the $C_n$ character tables from the theory of point groups. We make experimental predictions for the Bose-Einstein condensate, and show that, though these are the analog of some of the simplest problems in linear quantum mechanics, nonlinearity introduces surprising phenomena.

• Received 15 November 1999

DOI:https://doi.org/10.1103/PhysRevA.62.063611