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We consider the nonlinear excitation localized near the thin layer with nonlinear properties separated by different nonlinear media. The excitations are described by nonlinear Schrödinger equation (NLSE) with nonlinear potential. The problem is reduced to the solution of the NLSE with the boundary conditions of a special kind. We obtain the exact solutions of NLSEs satisfying the boundary conditions. We show that the existence of nonlinear localized excitations of four types is possible in a wide energy range. We derive the energy of localized excitations in the explicit form in the long-wave approximation. The conditions of localized state existence are found.

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In the analysis of dynamics of an ideal system as well as a system with point defects, the role of interaction is considered not only for the nearest neighbors. The Green’s function is constructed for steady-state vibrations of a chain at all possible frequencies. It is shown that, if the interaction with the next-to-nearest neighbors is taken into account, the Green’s function inevitably becomes double partial, the nature of its two components depending significantly on its eigenfrequency. It is found that the Green’s function for frequencies of the continuous spectrum of small vibrations has one component of the plane wave type, while the other component is localized near the source of perturbations. Such a Green’s function describes the so-called quasilocal vibrations. At certain discrete frequencies falling in the continuous spectrum, the quasilocal vibration is transformed into local vibration (that does not propagate to infinity). The conditions of applicability of differential equations with fourth spatial derivative are analyzed for describing the longwave vibrations of the atomic chain. Relations between parameters of atomic interactions permitting the use of such equations are formulated. Asymptotic forms of soliton fields in a nonlinear medium with spatial dispersion are discussed. It is shown that most of the soliton parameters are determined by the dispersion relation for the linearized equation.

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