An averaged evolution equation is presented and its dynamics studied for a mode-locked laser where the intensity discrimination (saturable absorption) in the cavity is provided by phase-sensitive amplification. The phase-sensitive amplifier acts as a phase-filter for selecting the specific intensity-dependent phase-rotation of the mode-locked pulse that locks the phase to the amplifier pump phase. The resulting averaged equation is a Swift-Hohenberg type model which is a fourth-order diffusion equation with cubic-quintic nonlinearities. Additionally, the governing evolution has a new linear growth term which couples to the nonlocal cavity energy. This parameter is a standard bifurcation parameter in Swift-Hohenberg models and is controlled by the cavity saturable gain. Such a modification to the governing evolution is the first of its kind to be considered theoretically in the context of the Swift-Hohenberg equation, and its significant impact on the mode-locked pulse dynamics and multi-pulsing behavior is explored.
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