Stationary solutions of the one-dimensional nonlinear Schrödinger equation. I. Case of repulsive nonlinearity

Abstract

All stationary solutions to the one-dimensional nonlinear Schr\"odinger equation under box and periodic boundary conditions are presented in analytic form. We consider the case of repulsive nonlinearity; in a companion paper we treat the attractive case. Our solutions take the form of stationary trains of dark or gray density-notch solitons. Real stationary states are in one-to-one correspondence with those of the linear Schr\"odinger equation. Complex stationary states are uniquely nonlinear, nodeless, and symmetry breaking. Our solutions apply to many physical contexts, including the Bose-Einstein condensate and optical pulses in fibers.