Localized states and their stability in an anharmonic medium with a nonlinear defect
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Localized States and Their Stability in an Anharmonic Medium with a Nonlinear Defect I. V. Gerasimchuk* Institute of Magnetism, National Academy of Sciences of Ukraine and the Ministry of Education and Science of Ukraine, Kiev, 03142 Ukraine National Technical University of Ukraine “Kyiv Polytechnic Institute,” Kiev, 03056 Ukraine *e-mail: [email protected] Received March 17, 2015
Abstract—A comprehensive analysis of soliton states localized near a plane defect (a defect layer) possessing nonlinear properties is carried out within a quasiclassical approach for different signs of nonlinearity of the medium and different characters of interaction of elementary excitations of the medium with the defect. A quantum interpretation is given to these nonlinear localized modes as a bound state of a large number of elementary excitations. The domains of existence of such states are determined, and their properties are analyzed as a function of the character of interaction of elementary excitations between each other and with the defect. A full analysis of the stability of all the localized states with respect to small perturbations of amplitude and phase is carried out analytically, and the frequency of small oscillations of the state localized on the defect is determined. DOI: 10.1134/S1063776115100076
1. INTRODUCTION The states localized near a defect (impurity) were comprehensively analyzed by Lifshits (see the surveys [1, 2]). Recently, the study of this type of states has been associated with the nonlinear properties of the medium or the defect. The problem of localization of excitations on an isolated defect is the first step in the study of spatial localization in periodic (modulated) systems with defects, first of all, in layered structures of various types (see, for example, [3–20]). These can be magnetic multilayer systems, which are of interest in view of the magneto-optical properties of a multilayer material, as well as the phenomena of giant magnetoresistance in them; layered crystals with multiatomic unit cell; and artificially layered semiconductors fabricated by multilayer deposition, which are widely used in superlattice electronic devices. In nonlinear optics, layered media are used in fiber-optic systems, optical delay lines, etc. (see, for example, [3, 5– 7, 9, 10, 14, 15]. In nonlinear optics, one usually considers a nonlinear medium that contains narrow layers with properties different from those of the medium. For stationary-profile waves, the problem is equivalent to the study of nonlinear excitations in a one-dimensional system with point defects (nonlinear local oscillations). For a single isolated linear defect, such a problem was analyzed in detail in [21–23] for different signs of nonlinearity of the medium and different characters of interaction between elementary exci-
tations and the defect. Systems with nonlinear defects in a linear medium were considered, for example, in [6, 23]. In the present work, we apply a nonlinear Schrödinger equation (NSE) with arbitrary sign of non
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