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Within the nonlinear Ginzburg-Landau theory, we study the superconducting state in a mesoscopic wire containing a narrow constriction in the presence of a uniform magnetic field directed along the wire. If the narrow region is small enough so that no vortices can penetrate through it, curved vortices are formed, i.e. they enter at the top of the sample (the widest part) and exit near the constriction. At high magnetic fields a giant vortex is nucleated in the widest part of the wire which breaks up into a smaller giant and/or individual vortices near the constriction.

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Within the time-dependent Ginzburg-Landau theory we discuss the effect of nonmagnetic interactions between the normal current and supercurrent in the presence of electric and magnetic fields. The correction due to the current-current interactions is shown to have a transient character so that it contributes only when a system evolves. Numerical studies for thin current-carrying superconducting strips with no magnetic feedback show that the effect of the normal current corrections is more pronounced in the resistive state where fast-moving kinematic vortices are formed. Simulations also reveal that the largest contribution due to current-current interactions appears near the sample edges, where the vortices reach their maximal velocity.