A modified Gross-Pitaevskii approximation was recently introduced for bosons in dimension d less than or equal to 2 by Kolomeisky et al (Kolomeisky E B, Newman T J, Straley J P and Qi X 2000 Phys. Rev. Lett. 85 1146). We use the density-functional approach with a sixth-degree interaction energy term in the Bose field to reproduce the stationary-frame results of Kolomeisky et al for a one-dimensional Bose-Einstein system with a repulsive interaction. We also find a soliton solution for an attractive interaction, which may be boosted to a finite velocity by a Galilean transformation. The stability of such a soliton is discussed analytically. We provide a general treatment of stationary solutions in one dimension which includes the above solutions as special cases. this treatment leads to a variety of stationary wave solutions for both attractive and repulsive interactions.