Abstract
We perform a systematic study of the linear stability and bifurcation structure of solitary wave, front, and periodic wave-train solutions which arise in the optical parametric oscillator near resonance detuning. In this limit, the dynamics is governed by a fourth-order quintic diffusion equation that describes the onset of spatial patterns beyond the equilibrium state. A complete analytic characterization is given of pulse and front solutions, which take a hyperbolic secant and hyperbolic tangent shape, respectively. Supporting computational evidence is given for the stability structure of such solutions. Periodic wave-train solutions of the Jacobi elliptic form are found and are explored numerically.
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