  Nonlinear wave formation and propagation on a complex network with excitable
node dynamics is of fundamental interest in diverse fields in science and
engineering. Here, we propose a new model of the Kuramoto type to study
nonlinear wave generation and propagation on circular subgraphs of a complex
network. On circle networks, in the continuum limit, this model is equivalent
to the over-damped Frenkel-Kontorova model. The new model is shown to keep the
essential features of those well-known models such as the diffusively coupled
B\"ar-Eiswirth model but with much simplified expression such that analytic
analysis becomes possible. We classify traveling wave solutions on circle
networks and show the universality of its features with perturbation analysis
and numerical computation.
