Under consideration in this paper is rogue waves on the general periodic traveling wave background of an integrable extended modified Korteweg–de Vries equation. The general periodic traveling wave solutions are presented. By means of the Darboux transformation and the nonlinearization of the Lax pair, we present the first‐, second‐ and third‐order rogue waves on the general periodic traveling wave background. Furthermore, the dynamic behaviors of rogue periodic waves are elucidated from the viewpoint of three‐dimensional structures.
The nonlocal nonlinear Gerdjikov-Ivanov (GI) equation is one of the most important integrable equations, which can be reduced from the third generic deformation of the derivative nonlinear Schrödinger equation. The Darboux transformation is a successful method in solving many nonlocal equations with the help of symbolic computation. As applications, we obtain the brightdark soliton, breather, rogue wave, kink, W-shaped soliton and periodic solutions of the nonlocal GI equation by constructing its 2n-fold Darboux transformation. These solutions show rich wave structures for selections of different parameters. In all these instances we practically show that these solutions have different properties than the ones for local case.
Analysis of the attenuation characteristics of environmental vibrations with different frequencies is of great benefit for preventing many vibration-related problems. This study carried out an in situ experimental study on the attenuation characteristics of a series of single frequency ground vibrations caused by harmonic excitations. An electromagnetic vibration excitation system was adopted to generate harmonic excitations at frequencies varying from 10 Hz to 80 Hz with steps 5 Hz. 5 low frequency, 1 component, 991B vibration sensors fastened on the ground surface were used for the measurement of the vertical velocity time histories of ground vibrations at 5 test points with different distances from the vibration source. The vibration level in terms of the vertical peak particle velocity of single frequency ground vibrations in general decayed monotonically but nonlinearly with distance, however, the attenuations of 15 Hz, 45 Hz, 65 Hz and 75 Hz ground vibrations were oscillatory due to vibration interference on the ground surface. The attenuation of single frequency ground vibrations was more rapid in the zone close to the vibration source than that in the zone far away. This study demonstrates that comparison of the attenuation characteristics of high and low frequency ground vibrations should take into account the difference in the amplitudes of the corresponding ground vibrations.
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