Complex systems of Kuramoto–sine-Gordon solitons

Abstract

The 1 + 1 dimensional Kuramoto–sine-Gordon system consists of a set of N nonlinear coupled equations for N scalar fields θ i , which constitute the nodes of a complex system. These scalar fields interact by means of Kuramoto nonlinearities over a network of connections determined by N(N − 1)/2 symmetric coupling coefficients a ij . This system, regarded as a chirally invariant quantum field theory, describes a single decoupled massless field together with N − 1 scalar boson excitations of nonzero mass depending on a ij , which propagate and interact over the network. For N = 2 the equations decouple into separate sine-Gordon and wave equations. The system allows an extensive array of soliton configurations which interpolate between the various minima of the 2π-periodic potential, including sine-Gordon solitons in both static and time-dependent form, as well as double sine-Gordon solitons which can be imbedded into the system for any N. The precise form of the stable soliton depends critically on the coupling coefficients a ij . We investigate specific configurations for N = 3 by classifying all possible potentials, and use the symmetries of the system to construct static solitons in both exact and numerical form.