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arXiv:cond-mat/0109235 · Physics Letters A

Exact form of the Bogoliubov excitations in one-dimensional nonlinear Schrödinger equation

Dmitry L. Kovrizhin

Published Sep 1, 2001• 49 citations
arXivHTMLPDF
DOI
10.1016/s0375-9601(01)00503-5

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In this short note, we construct mappings from one-dimensional integrable spinor BECs to matrix nonlinear Schrödinger equation, and solve the Bogoliubov equation of these systems. A map of spin-nnn BEC is constructed from the 2n2^n2n-dimensional spinor representation of irreducible tensor operators of so(2n+1)so(2n+1)so(2n+1). Solutions of Bogoliubov equation are obtained with the aid of the theory of squared Jost functions.
We study relaxation dynamics in one-dimensional Bose gases, formulated as an initial value problem for the classical nonlinear Schr\"odinger equation. We propose an analytic technique which takes into account the exact spectrum of nonlinear 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 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.
Using a time-dependent modified nonlinear Schrödinger equation (MNLSE)-where the conventional chemical potential proportional to the density is replaced by the one inferred from Lieb-Liniger's exact solution-we study frequencies of the collective monopole excitations of a one-dimensional Bose gas. We find that our method accurately reproduces the results of a recent experimental study [E. Haller et al., Science 325, 1224 (2009)] in the full spectrum of interaction regimes from the ideal gas, through the mean-field regime, through the mean-field Thomas-Fermi regime, all the way to the Tonks-Giradeau gas. While the former two are accessible by the standard time-dependent NLSE and inaccessible by the time-dependent local density approximation, the situation reverses in the latter case. However, the MNLSE is shown to treat all these regimes within a single numerical method.