On the Marchenko system and the long-time behavior of multi-soliton solutions of the one-dimensional Gross-Pitaevskii equation

TL;DR

This paper rigorously analyzes the Marchenko system related to the 1D Gross-Pitaevskii equation, establishing well-posedness, constructing multi-soliton solutions, and studying their long-time asymptotic behavior.

Contribution

It provides the first rigorous well-posedness results for the Marchenko system associated with the GP equation and constructs multi-soliton solutions with asymptotic analysis.

Findings

Established well-posedness of the Marchenko system for GP

Constructed multi-soliton solutions as superpositions of traveling waves

Analyzed the long-time asymptotic behavior of solutions

Abstract

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.