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A sample including 935 blazars with 415 Flat Spectrum Radio Quasars (FSRQs) and 520 BL Lac objects (BL Lacs) (233 HBLs (High synchrotron peak frequency BL Lacs), 144 IBLs (Intermediate synchrotron peak frequency BL Lacs), and 143 LBLs (Low synchrotron peak frequency BL Lacs)) is selected in Fermi 3FGL catalogue. The correlations between two flux densities of the radio 1.4 GHz and γ -ray at 0.1, 0.3, 1, 3, and 10 GeV are investigated for the samples of all blazars, FSRQs, BL Lacs, and subclasses of HBLs and LBLs, respectively. Results show that there are strong positive correlations between the two flux densities, with the correlation coefficients r = 0.48 ∼ 0.81 , and the chance probabilities p all less than 10 − 4 . The correlation coefficient has different trends with the change of γ -ray emission frequency for different blazar samples, and the average correlation coefficients for all the samples at five respective bands decrease with the increase of γ -ray frequency. Therefore, with the increase of γ -ray frequency, the dominant mechanism of γ -ray radiation of blazar is changing, and at the same frequency the dominant mechanism is different for different samples. The γ -ray emission of HBLs may be mainly dominated by the synchrotron self-Compton, while the other mechanisms are responsible for the γ -ray emissions of LBLs. The origin of γ -ray in FSRQs is more complex than that of BL Lacs.

In this paper, the flux densities at the radio (R) 1.4 GHz, optical (O) 4.68 × 10 14 Hz, X-ray (X) 1 keV, and γ -ray ( γ ) 1 GeV are obtained from the available literature for a Fermi blazar sample. The 6 effective spectral indices ( α RO , α RX , α R γ , α OX , α O γ , and α X γ ) between any two of above bands are calculated. The correlations between any two spectral indices are investigated for the whole blazar sample and the subclass samples of FSRQs (Flat Spectral Radio Quasars) and BL Lacs including High synchrotron peak frequency BL Lacs (HBLs) and Low synchrotron peak frequency BL Lacs (LBLs). Results show that: (1) There are close correlations between any two spectral indices for all samples, except for the sample of HBLs and the correlation between α OX and α R γ . This kind of correlations can be explained by the structures of energy spectral distributions at four bands. (2) In the scatter plots of effective spectral index correlations, FSRQs and LBLs are distributed in the same area, but HBLs and FSRQs (LBLs) are distributed in the different areas. (3) The separation degree (discrimination) between the distributions of FSRQs (LBLs) and HBLs differs with the scatter plot of effective spectral index correlations, and the discrimination is related to the frequencies to determine two effective spectral indices.

The X-ray emissions of blazars are located at the end of synchrotron radiation and the beginning of inverse Compton radiation. Therefore, the origin of the X-ray emissions is rather complex. The spectral energy distributions (SEDs) of blazars from radio to X-ray bands can be fitted approximatively by a parabolic function. If we consider approximately the fitting curves as the physical spectra of blazars to analyze the X-ray emissions of Fermi blazars, the results show that: (1) The X-ray emissions of blazars contain two components, i.e. the synchrotron radiation and inverse Compton radiation, which can be simply separated by these fitting curves; (2) the higher the synchrotron peak frequency of the source, the greater the synchrotron radiation component, and the less the inverse Compton radiation component; (3) at 1 keV of the X-ray waveband, the synchrotron radiation component accounts for 17%, 27%, and 73% of the total X-ray emission, for FSRQs (Flat Spectrum Radio Quasars), LBLs (Low synchrotron peak frequency BL Lac objects), and HBLs (High synchrotron peak frequency BL Lac objects), respectively; (4) there is a strong positive correlation between the synchrotron peak frequency and the synchrotron radiation flux density at 1 keV, while no correlation exists between the synchrotron peak frequency and the inverse Compton radiation flux density; (5) the radiation mechanism of LBLs may be similar to that of FSRQs in the X-ray waveband.

We have developed a systematic analytical approach to the study on the dynamic properties of the linear and the nonlinear excitations for quasi-one-dimensional Bose–Einstein condensate trapped in optical lattices. A novel linear dispersion relation and an algebraic soliton solution of the condensate are derived analytically under consideration of Bose–Einstein condensate with a periodic potential. By analysing the soliton solution, we find that the interatomic interaction strength has an important effect on soliton dynamic properties of Bose–Einstein condensate.