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Optical resonators are becoming increasingly important due to their nonlinear properties and potentially small sizes. In recent years, it was discovered that these structures are able to support different kinds of localized solutions depending on the competition between the group velocity dispersion and higher-order nonlinearities. However, as the resonators become smaller, the number of modes decreases until the group velocity dispersion effect can be ignored. Here, we study the dynamics of two symmetric optical modes coupled in a single-mode ring resonator. We demonstrate the existence of localized pulse and kink solutions in this set-up, owing to the intrinsic Kerr nonlinearity. We show that pulse solutions lie on the boundary between different oscillating regimes in phase space, while kink solutions are found at the transition from symmetric to asymmetric pulse solutions. When perturbing the localized solutions, they undergo critical bifurcations to form persistent periodic and chaotic oscillations, some of which exhibit self-switching behaviours. We also show that these localized solutions are the organizing centre of various dynamical regimes in the cavity, including spontaneous symmetry breaking, optical bi-stability and multi-stability of homogeneous states, and Hopf bifurcation. We discuss how the ratio between the self- and cross-phase modulation parameters influences these dynamical regimes and, therefore, the existence of pulse and kink solutions. We use analytical techniques and numerical continuation methods to compute the loci of all these dynamical behaviours and present our results as comprehensive phase diagrams in the parameter plane. Overall, our work showcases the distinct and exotic behaviour that should be expected from this optical device.
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