Theory of interaction and bound states of spiral waves in oscillatory media

Vortex Glass and Vortex Liquid in Oscillatory Media

“…k ≃ −c −1 exp(−π/|2c|) for b = 0 and |c| ≪ 1 [10]. In the parameter region where the singlespiral solution is stable and c = b, the interaction between two well-separated spirals falls off exponentially [11][12][13]. The screening was attributed to the shock lines where the waves emitted by the cores collide.…”

“…The screening was attributed to the shock lines where the waves emitted by the cores collide. For relatively small b, c the interaction is monotonic, and the spirals exhibit asymptotic repulsion irrespectively of their charge ("monotonic range") [11,12]. In contrast, for larger b, c satisfying the condition (c − b)/(1 + bc) > c * ≈ 0.845 the velocity vs distance dependence is modulated ("oscillatory range"), and the spirals become keen to form a variety of long-living (but unstable) bound states [11,14].…”

“…For relatively small b, c the interaction is monotonic, and the spirals exhibit asymptotic repulsion irrespectively of their charge ("monotonic range") [11,12]. In contrast, for larger b, c satisfying the condition (c − b)/(1 + bc) > c * ≈ 0.845 the velocity vs distance dependence is modulated ("oscillatory range"), and the spirals become keen to form a variety of long-living (but unstable) bound states [11,14]. While the above interactions cannot account for the strongly chaotic regimes where many defects are spontaneously generated and undergo violent motion, they are expected to play a leading role in the occurrence of the quasi-static structures of large and small spirals surrounded by a complex network of shocks commonly observed in the large region of the (b, c) parameter space where the single spiral solution is stable [2][3][4].…”

“…A distinctive feature of the two-dimensional CGLE is the existence of nontrivial sources of spiral waves (vortices) which determine the oscillating frequency of the entire system [9]. The single-spiral solution for a vortex centered at r 0 reads as: [10,11]:…”

“…A distinctive feature of the two-dimensional CGLE is the existence of nontrivial sources of spiral waves (vortices) which determine the oscillating frequency of the entire system [9]. The single-spiral solution for a vortex centered at r 0 reads as: [10,11]:where r = |r − r 0 |, θ is the polar angle measured from the vortex core, φ is an arbitrary phase, and m = ±1 is the topological charge. Far away from the core, the solution approaches a plane wave with ψ(r) ≃ kr, where the asymptotic wavenumber k is related to the rotation frequency as ω = −c − (b − c)k 2 .…”