Strunin, D. V. (2009) Phase equation with nonlinear excitation for nonlocally coupled oscillators. Physica D: Nonlinear Phenomena, 238 (18). pp. 19091916. ISSN 01672789

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Abstract
Some reactiondiffusion systems feature nonlocal interaction and, near the point of Hopf bifurcation, can be represented as a system of nonlocally coupled oscillators. Phase of oscillations satisfies an evolution pde which takes different forms depending on the values of parameters. In the simplest case the equation is
effectively a diffusion equation which is excitationfree. However, more complex forms are possible such as the Nikolaevskii equation and the Kuramoto–Sivashinsky equation incorporating linear excitation. We analyse a situation when the phase equation is based on nonlinear excitation. We derive conditions on the values of the parameters leading to the situation and show that the values satisfying the conditions exist.
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