| Summary: | We discuss the Kerr nonlinearities of the nonlocally nonlinear system with oscillatory responses by the variational approach. The self-focusing and self-defocusing states are found to dramatically depend on the degree of nonlocality. When the degree of nonlocality goes across a critical value, the nonlinearity can transit to its opposite counterpart, that is, focusing to defocusing or defocusing to focusing. The critical degree of nonlocality for the nonlinearities transition is given analytically, and confirmed by numerical simulations. As a versatile mathematical tool, we also employ the variational approach to analytically address the stabilities of solitons, and obtain the range of the degree of nonlocality for the stable solitons, which is confirmed by the linear stability analysis and the numerical simulation.
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