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We study properties of autowave patterns in reaction–diffusion media with non-uniform chiral anisotropy of the diffusion tensor. The curvature–velocity relation for circular wave fronts in such media is derived. This relation is confirmed by numerical simulations. Strong dependence of spiral waves angular velocity on the angle of chirality is predicted theoretically and confirmed numerically. Long-ranged interaction between spiral waves is predicted and observed in computer experiments.

Propagation of ring-shaped autowaves on deformed cylindrical surfaces is studied theoretically within a kinematical model. Theoretical results predict the existence of a critical deformation of the surface above which propagation of ring-shaped fronts is impossible and the structure of stable autowaves becomes similar to the so-called V-patterns with free ends. This prediction is confirmed by numerical simulations of a two-variable reaction–diffusion excitable medium on a torus.

An explicit expression for the velocity of propagation of ring-shaped autowaves on arbitrary surfaces of revolution is derived. It is shown that under certain conditions the surface can serve as a diode for autowaves. Theoretical predictions are confirmed by computer simulations.

The weakly nonlinear dynamics of surface magnetoacoustic waves in magnetic flux tubes in the presence of the effects of dissipation and active non-adiabaticity is considered. The wave is described either by the extended Benjamin-Ono equation (slab geometry) or by the extended Leibovich-Roberts equation (cylindrical geometry). For the slab geometry, it is shown that solitary waves with arbitrary initial parameters evolve to solitons with specific amplitude, length and speed defined by the physical parameters of the magnetic slab.

An autowave model of the spiral pattern formation in magnets located in a rotating magnetic field is proposed. The model is based on the overdamped double sine-Gordon equation. Nucleation of spiral domains is associated with the averaged motion of domain walls in the rotating field. A vortex-type defect (Bloch line) is the core of the spiral.

New non-spiral propagating autowave structures in unrestricted excitable media are predicted theoretically. The front shape of the new structures is evaluated analytically. The structures are shown to be stable with respect to small wave front perturbations.

The effect on an autowave vortex of a nonlocalized, periodic external forcing with a near-resonant frequency is examined in a bounded medium. The periodic forcing can produce movement of the vortex and such a resonantly drifting vortex can be repelled from a boundary.

Autowave propagation is studied in a bistable, piecewise linearized Schlogl model of an autocatalytic process across an area (gap) with inhibited autocatalysis. Exact analytical results on the time-dependence of this process are derived. Furthermore, the possibility of autowave propagation on a catalyst surface with passivated regions is examined.

A two-dimensional waves classification based on the topology of the wave-vector distribution is suggested. The conservation laws for two topological quantum numbers impose restrictions on the possible wave transformations. The predictions are in full agreement with observations.