Stability and bifurcation of spatially coherent solutions of the damped-driven NLS equation

An analytical study is conducted of the structure, stability, and bifurcation of the spatially dependent time-periodic solutions of the damped-driven sine-Gordon equation in the nonlinear Schrodinger approximation. Locked states are found for which the spatial structure consists of coherent excitations localized about x = 0 or L/2. A bifurcation analysis reveals the relationship of these spatially localized solutions to the spatially independent ones and provides a cutoff wavenumber above which there are no spatially dependent solutions; this establishes an upper bound on the number of local excitations comprising the spatial pattern. A linear stability analysis shows that the spatially localized solutions undergo a Hopf bifurcation to temporal quasi-periodicity as the driver amplitude Γ is increased. For sufficiently high driver frequencies, the temporally periodic solution regains its stability (via another Hopf bifurcation) in a Γ-window of finite width before undergoing a third Hopf bifurcation to quasi-periodicity. The analytical results compare favorably with numerical solutions and provide the requisite ingredients for construction of chaotic attractors for this system.