This brief review recalls some chapters in theory of sliding incommensurate density waves which may have appeared after inspirations from studies of I.E Dzyaloshinskii and collaborations with him. First we address the spin density waves which rich order parameter allows for an unusual object of a complex topological nature: a half-integer dislocation combined with a semi-vortex of the staggered magnetization. It becomes energetically preferable with respect to an ordinary dislocation due to the high Coulomb energy at low concentration of carriers. Generation of these objects should form a sequence of pi-phase slips in accordance with experimental doubling of phase-slips rate. Next, we revise the commonly employed TDGL approach which is shown to suffer from a violation of the charge conservation law resulting in nonphysical generation of particles which is particularly pronounced for electronic vortices in the course of their nucleation or motion. The suggested consistent theory exploits the chiral transformations taking into account the principle contribution of the fermionic chiral anomaly to the effective action. The derived equations clarify partitions of charges,currents and rigidity among subsystems of the condensate and normal carriers and the gluing electric field. Being non-analytical with respect to the order parameter, contrarily the conventional TDGL type, the resulting equations still allow for a numerical modeling of transient processes related to space- and spatio-temporal vorticity in DWs.
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