  We establish a rigorous well-posedness results for the Marchenko system
associated to the scattering theory of the one dimensional Gross-Pitaevskii
equation (GP). Under some assumptions on the scattering data, these
well-posedness results provide regular solutions for (GP). We also construct
particular solutions, called $N-$soliton solutions as an approximate
superposition of traveling waves. A study for the asymptotic behaviors of such
solutions when $t\rightarrow\pm \infty$ is also made.
