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2026
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In this paper, we consider the focusing fractional nonlinear Schrödinger equation (FNLS) on the waveguide manifolds [Formula: see text], both in the isotropic and anisotropic cases. Under different conditions, we establish the existence and periodic dependence of the ground states of the focusing FNLS. In the intercritical regime, we also establish the large data scattering for the anisotropic focusing FNLS by utilizing the framework of semivirial vanishing geometry.