We study the one-dimensional Gross-Pitaevskii equation, a cubic defocusing non-linear Schrodinger equation with nonvanishing boundary conditions. In particular we linearize around the dark solitons, which are a family of exact solutions that do not decay at spatial infinity (as opposed to bright solitons in the focusing NLS). The dark solitons we study are exact solutions of the Gross-Pitaevskii equation, which has been shown to be completely integrable by means of the inverse scattering transform. In particular we linearize around these solitons to produce specific matrix operators which contain important spectral data. Such information is understood by discovering the main ingredients to build up distorted Fourier transforms and projections on to eigenvalues. These are the Jost functions, namely bounded solutions to the eigenvalue problem, which are then used to compute the resolvent kernel. We give a comprehensive description of the long-time dynamics exhibited by perturbations of the black soliton after looking carefully at the vacuum steady state case (the ‘whitest’ dark soliton 1). This is very informative as the constant coefficient problem has many similarities to the more generic black soliton case. In particular they both give rise to singular behavior at zero energy. So when studying the evolution of the perturbation we observe special asymptotics at low frequencies. Finally we derive the scattering theory for the matrix differential operator in the more general gray soliton case. This is motivated by the fact that understanding certain properties of this operator can lead to a new proof for orbital stability.
Malik, Numann,
"Dark soliton linearization of the 1D Gross-Pitaevskii equation"
(2018).
Mathematics Theses and Dissertations.
Brown Digital Repository. Brown University Library.
https://doi.org/10.26300/3fkc-be61
