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Using a time-dependent modified nonlinear Schr\"odinger equation (m-NLSE) --where the conventional chemical potential proportional to the density isreplaced by the one inferred from Lieb-Liniger's exact solution -- we studyfrequencies of the collective monopole excitations of a one-dimensional (1D)Bose gas. We find that our method accurately reproduces the results of a recentexperimental study [E. Haller et al., Science Vol. 325, 1224 (2009)] in thefull spectrum of interaction regimes from the ideal gas, through the mean-fieldregime, through the mean-field Thomas-Fermi regime, all the way to theTonks-Giradeau gas. While the former two are accessible by the standardtime-dependent NLSE and inaccessible by the time-dependent local densityapproximation (LDA), the situation reverses in the latter case. However, them-NLSE treats all these regimes within a single numerical method.

Monopole excitations of a harmonically trapped one-dimensional Bose gas from the ideal gas to the Tonks-Girardeau regime