On the Marchenko System and the Long-time Behavior of Multi-soliton Solutions of the One-dimensional Gross-Pitaevskii Equation (Journal of Partial Differential Equations, 2015)
" 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 Nsoliton solutions as an approximate superposition of traveling waves. A study for the asymptotic behaviors of such solutions when t \u2192 \u00b1 \u221e is also made."
Keywords: Non-linear Schrödinger equation; Gross-Pitaevskii equation