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We study the existence of one-dimensional localized states supported bylinear periodic potentials and a domain-wall-like Kerr nonlinearity. The modelgives rise to several new types of asymmetric localized states, includingsingle- and double-hump soliton profiles, and multihump structures. Exploitingthe linear stability analysis and direct simulations, we prove that theselocalized states are exceptional stable in the respective finite band gaps. Themodel applies to Bose-Einstein condensates loaded onto optical lattices, and inoptics with period potentials, e.g., the photonic crystals and opticalwaveguide arrays, thereby the predicted solutions can be implemented in thestate-of-the-art experiments.

Asymmetric localized states in periodic potentials with a domain wall-like Kerr nonlinearity