Solitons in complex systems of chiral fields with Kuramoto interactions

We construct a complex system of N chiral fields, each regarded as a node or a constituent of a complex field-theoretic system, which interact by means of chirally invariant potentials across a network of connections. In the classical case, these interactions are identical or similar to Kuramoto interactions, leading to synchronization phenomena for the well-known Kuramoto model and its many extensions and generalizations to higher dimensions. We consider chiral systems of arbitrary size N, where each constituent carries a conserved charge of topological origin, which evolve according to a coupled system of second-order, Lorentz invariant, nonlinear partial differential equations. Stable soliton configurations occur as a consequence of the nonlinear network interactions, not necessarily from self-interactions of the fundamental fields. In 1+1 dimensions, these chirally invariant models allow for multi-soliton configurations that for N=2 are determined by the sine-Gordon equation and for N=3 reduce in special cases to the double sine-Gordon equation, which has exact double-kink static solutions consisting of solitons positioned at arbitrary locations. Planar and three-dimensional networked skyrmions appear in higher dimensions. Such configurations can be viewed for general N as bound states of the constituent fields, which exist together with the usual fundamental excitations. Whereas Kuramoto interactions in first-order systems lead to emergent classical phenomena such as synchronization, these same interactions in complex systems of chiral fields result in a rich variety of multi-soliton bound states.