For studying the vortex structure in uniform dense dusty astrophysical conditions, a two-dimensional nonlinear equation is derived employing the quantum magnetoplasma hydrodynamic model and considering the strong collisional effect. The coherent vortex solution is obtained by perturbation analysis method. It is shown that the distribution of the electrostatic potential forms spatially a periodic vortex street, and is controlled temporally by the arbitrary function of time that may lead to abundant spacial distributions. It is found that the dust charge number, collision frequency, electron Fermi wavelength and quantum correction all play significant roles to the spatial distribution of vortex street.
For a special coupled mKdV system, which can be derived from a two-layer fluid model, Hirota's bilinear direct method is used to construct and yield the complexiton solutions. The detailed physical properties of complexitons are further illustrated graphically.
With Hirota's bilinear direct method, we study the special coupled KdV system to obtain its new soliton solutions. Then we further discuss soliton evolution, corresponding structures, and interesting interactive phenomena in detail with plot. As a result, we find that after the interaction, the solitons make elastic collision and there are no exchanges of their physical quantities including energy, velocity and shape except the phase shift.
For an inhomogeneous quantum magnetoplasma system with density and temperature gradients, a two-dimensional nonlinear fluid dynamic equation is derived in the case where the collision frequency between ions and neutrals is minor. The shock, explosion and vortex solutions of the potential for this system are obtained. The changes of the potential in the dense astrophysical environment are discussed. It is shown that the strength of the shock and the width of the explosion are both enhanced with the density increasing (equivalently, the normalized quantum parameter decreasing), but with the drift velocity decreasing (equivalently, the density and temperature gradients decreasing); the potential always tends to a stable value with the spatiotemporal phase increasing, and the system approaches finally to a stable state. Besides, the temporal and spatial distributions of the vortex potential display a stable and period vortex street.
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