Forced vibrations and resonance wave scattering on impurity in 1D discrete lattice with nearest- and next-nearest-neighbors interaction

Kosevich, A. M.; Савотченко, С. Е.

doi:10.1016/s0921-4526(99)02813-6

Cited by 10 publications

(5 citation statements)

References 4 publications

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“…(5), and its parameters specified by Eqs. (10) and (11) are transformed into the corresponding parameters given by Eq. (6).…”

Section: Russian Physicsmentioning

confidence: 99%

“…This parameter arises if we consider interactions with the nearest and more distant neighbors in the crystal lattice. This approach was first examined in [9,10].With allowance for the foregoing, we now include terms describing the potential energy of interactions with the second neighbors into the Hamiltonian. Then the model Hamiltonian of the examined system assumes the form 2 2 2 1 1 2 2…”

mentioning

confidence: 99%

“…This parameter arises if we consider interactions with the nearest and more distant neighbors in the crystal lattice. This approach was first examined in [9,10].…”

mentioning

confidence: 99%

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Nonlinear Collective Excitations in Quasi—One-Dimensional Structures with Spatial Dispersion

Савотченко

2005

Russ Phys J

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It is demonstrated that a description of the effects of higher-order dispersion of continuous oscillations calls for a consideration of interactions with the nearest and more distant neighbors in a crystal lattice. Exact soliton and periodic solutions of the nonlinear equation with fourth-order derivatives describing long-wavelength collective excitations in a quasi-one-dimensional chain have been found.Interest in the study of special features of nonlinear excitation propagation in systems with dispersion has quickened in the past few years [1-3]. These investigations find application, for example, in the theory of structural phase transitions in ferroelectrics caused by displacements of the equilibrium positions of atoms [4][5][6][7]. Frequencies of oscillatory modes in these systems decrease sharply when the temperature of the system approaches the critical one. As a result, ordered local regions arise that manifest themselves as additional peaks in neutron scattering [8].A theoretical description of such systems is based on the model of quasi-one-dimensional chain of identical oscillators [4]. The model suggests that each oscillator corresponds to a particle of mass m moving in a symmetric external potential field. Below we consider two sublattices, with heavy atoms of the first sublattice spaced at distance a and light atoms of mass m of the second sublattice displaced by u n from the heavy atoms.As is well known, the long-wavelength limit in the description of mechanics of a discrete system based on the use of differential equations with the second-order spatial derivatives remains unchanged for arbitrary number of interacting neighbors. Therefore, this approximation can be developed on the basis of the model considering interactions with the nearest neighbors with inclusion of the interparticle interaction parameter. If higher-order dispersion is considered, it will be described by an independent parameter. This parameter arises if we consider interactions with the nearest and more distant neighbors in the crystal lattice. This approach was first examined in [9,10].With allowance for the foregoing, we now include terms describing the potential energy of interactions with the second neighbors into the Hamiltonian. Then the model Hamiltonian of the examined system assumes the form 2 2 2 1 1 2 2

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“…(5), and its parameters specified by Eqs. (10) and (11) are transformed into the corresponding parameters given by Eq. (6).…”

Section: Russian Physicsmentioning

confidence: 99%

mentioning

confidence: 99%

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Nonlinear Collective Excitations in Quasi—One-Dimensional Structures with Spatial Dispersion

Савотченко

2005

Russ Phys J

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“…Локализованные вблизи дефектов малоамплитудные нелинейные колебания в ангармонических кристаллах при наличии пространственной дисперсии нелинейной среды используется НУШ с производными старших порядков [7]. Различные особенности динамики одномерных дискретных систем со взаимодействием не только ближайших соседей и роль высшей дисперсии в солитонной динамике рассматривались в работах [8][9][10][11][12][13]. Локализация на границе раздела линейной и нелинейной сред с учетом конечной интенсивностью взаимодействия возбуждения с плоским дефектом описана в [14].…”

Section: Introductionunclassified

" Nonlinear waves in a unlimited environment "

Савотченко¹

2018

NV BelSU Scientific bulletin Mathematics Physics

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АннотацияРассмотрены вопросы существования периодических стационарных возбуждений в полуограниченных ангармонических кристаллах с различными знаками нелинейности. Предложена модель, математическая формулировка которой представляет собой одномерную краевую задачу для нелинейного уравнения Шредингера на полуоси. В рассматриваемой системе в зависимости от значения частоты получены несколько типов стационарных состояний, описывающих периодические распределения поля возбуждения. Показано, что в средах с отрицательной нелинейностью существует один вид периодически распределенных состояний, а в средах с положительной нелинейностью -два вида. Такие состояния описываются периодическими решениями НУШ, содержащими эллиптические функции. Получены выражения, определяющие значения частот всех рассматриваемых состояний, в явном аналитическом виде и определены условия их существования.

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“…. Such equations can occur in describing long wave oscillations of discrete crystal lattices taking the nearest neighbors into account [5,6]. In this work, it is represented considering peculiarities of propagating nonlinear excitations in a hydrogen bond molecular chain with interaction between the nearest and second neighbor particles.…”

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confidence: 99%

Peculiarities of soliton motion in molecular systems with high dispersion

Krasilnikov¹,

Савотченко

2004

phys. stat. sol. (c)

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In this work, features of propagating protons along molecular chain of hydrogen bonds are described from position of soliton dynamics with taking into account interaction of first and second neighbors of a proton sublattice. It is proposed extension of the model that is an endless chain of water molecules in which formation of hydrogen bonds is due to participating one proton of every water molecule, a second proton no participating in hydrogen bond and being confined by covalent bond of an oxygen atom. Nonlinearity is due to peculiar properties of proton sublattice potential. The model used to obtain continual equations which contain the spatial derivatives of the fourth order that is related with dispersion of longwave oscillations. The availability of such a dispersion changes essentially dynamics of the molecular chain, which allows of manifesting new peculiarities of propagating nonlinear excitations. It is shown there are two new sorts of charge density excitations transferred by solitons determined as exact analytic dependences in such a system. . Such equations can occur in describing long wave oscillations of discrete crystal lattices taking the nearest neighbors into account [5,6]. In this work, it is represented considering peculiarities of propagating nonlinear excitations in a hydrogen bond molecular chain with interaction between the nearest and second neighbor particles. Due to this interaction, the derivatives of higher order are in the continual equations of motion. The presence of such derivatives of the order exceeding the second one causes higher dispersion effects having new peculiarities of propagating the longwave excitations in the chain.It is well known a proton conductivity is higher along hydrogen bond one-dimensional chains in certain crystal phase compounds (hard spirits, carbohydrates) by several orders than in a cross direction [7]. Besides it is known proton mobility in ice crystals is lower by only the one order than electron mobility in metals [8][9][10][11]. The basic proposition of the ice proton conductivity theory is a proton can be transferred along the chain as two ion defects: hydroxonium ion H 3 O + and hydroxyl ion OH − created by water molecule dissociation due to transferring one of its protons to a neighbor molecule according to the reaction scheme: 2H 2 O∆OH − +H 3 O

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Forced vibrations and resonance wave scattering on impurity in 1D discrete lattice with nearest- and next-nearest-neighbors interaction

Cited by 10 publications

References 4 publications

Nonlinear Collective Excitations in Quasi—One-Dimensional Structures with Spatial Dispersion

Nonlinear Collective Excitations in Quasi—One-Dimensional Structures with Spatial Dispersion

" Nonlinear waves in a unlimited environment "

Peculiarities of soliton motion in molecular systems with high dispersion

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