Interplay of Solitons and Radiation in One-Dimensional Bose Gases

We study relaxation dynamics in one-dimensional Bose gases, formulated as an initial value problem for the classical non-linear Schr\"{o}dinger equation. We propose an analytic technique which takes into account the exact spectrum of non-linear modes, that is both soliton excitations and dispersive continuum of radiation modes. Our method relies on the exact large-time asymptotics and uses the so-called dressing transformation to account for the solitons. The obtained results are quantitatively compared with the predictions of the linearized approach in the framework of the Bogoliubov--de Gennes theory. In the attractive regime, the interplay between solitons and radiation yields a damped oscillatory motion of the profile which resembles breathing. For the repulsive interaction, the solitons are confined in the sound cone region separated from the supersonic radiation.