Integrable Reductions of Manycomponent Magnetic Systems in (1,1) Dimensions
TL;DR: In this article, a generalized many component Heisenberg spin chain with phonon interaction is proposed, which can be reduced to different real magnetic systems such as many chained magnetic crystals with nontrivial interchain couplings, a mixture of many chained ferro and antiferromagnets, etc.
Abstract: A generalized many component Heisenberg spin chain with phonon interaction is proposed Some reductions of the proposed model leading to different real magnetic systems such as many chained magnetic crystals with nontrivial interchain couplings, a mixture of many chained ferro and antiferromagnets, a "colour" generalized Pierels-Hubbard model, etc, are studied It has been shown that the dynamics of all the above real models are close to some integrable systems and coincide with them in certain limits Such integrable systems are the coupled generalised system of Yajima and Oikawa and U(p, q) nonlinear Schrodinger equation, already well studied
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TL;DR: In this article, the relation between magnon-magnon interaction in the non-linear Schrodinger equation and phonon-phonon interaction is analyzed, and the authors discuss the localized quantum condensation of magnons on a one-dimensional magnetic chain.
5 citations
TL;DR: In this paper, the Cauchy problem for repulsive vector nonlinear Schrodinger equation under nonvanishing boundary conditions is studied via the inverse transform, and exact N-soliton solutions are constructed.
Abstract: The Cauchy problem for repulsive vector nonlinear Schrodinger equation under nonvanishing boundary conditions is studied via the inverse transform. For reflectionless potentials exact N-soliton solutions is constructed. The single soliton solution stability is proved as well under small perturbations of continuous spectrum. The perturbed soliton is shown to tend to a pure one asymptotically as 1/t.
3 citations
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TL;DR: In this paper, a generalization of the Toda lattice is proposed, where the original Flaschka-Manakov variables are coupled to newly introduced dependent variables, and the general case wherein the additional dependent variables are vector-valued is considered.
Abstract: We propose a new integrable generalization of the Toda lattice wherein the original Flaschka-Manakov variables are coupled to newly introduced dependent variables; the general case wherein the additional dependent variables are vector-valued is considered. This generalization admits a Lax pair based on an extension of the Jacobi operator, an infinite number of conservation laws and, in a special case, a simple Hamiltonian structure. In fact, the second flow of this generalized Toda hierarchy reduces to the usual Toda lattice when the additional dependent variables vanish; the first flow of the hierarchy reduces to a long wave-short wave interaction model, known as the Yajima-Oikawa system, in a suitable continuous limit. This integrable discretization of the Yajima-Oikawa system is essentially different from the discrete Yajima-Oikawa system proposed in arXiv:1509.06996 (also see this https URL) and studied in arXiv:1804.10224. Two integrable discretizations of the nonlinear Schrodinger hierarchy, the Ablowitz-Ladik hierarchy and the Konopelchenko-Chudnovsky hierarchy, are contained in the generalized Toda hierarchy as special cases.
2 citations
TL;DR: In this article, the SU(2) generalized coherent states technique was used to obtain in continual limit the classical description of the model, which is a system of coupled Landau-Lifshitz and Boussinesq equations describing nonlinear spin waves accompanied with sound waves.
Abstract: We investigate the Heisenberg magnetic chain of spins interacting with the atomic displacement by exchange interaction modulation. By use of the SU(2) generalized coherent states technique we obtain in continual limit the classical description of the model, which is a system of coupled Landau–Lifshitz and Boussinesq equations describing nonlinear spin waves accompanied with sound waves. Domain-walls and soliton type solutions of this system are derived in harmonic phonon approximation. The possibility of energy pumping from the magnon subsystem to the phonon one is shown for the case of near-sonic velocities of motion of the solitons. The model of the multisublattice Heisenberg magnet consisting of p sublattices of ferromagnetic type and q sublattices of antiferromagnetic type is studied taking into consideration the interaction between the sublattices. A system of equations describing this model, which in the small amplitude of spin deviations approximation can be reduced to the nonlinear Schrodinger equation with Yajima–Oikawa potential, is obtained.
2 citations
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TL;DR: In this paper, a Lax-pair representation for the discrete Yajima-Oikawa system was proposed, and the next higher flow of the discrete Schrodinger hierarchy was derived.
Abstract: A space discretization of an integrable long wave-short wave interaction model, called the Yajima-Oikawa system, was proposed in the recent paper arXiv:1509.06996 using the Hirota bilinear method (see also this https URL). In this paper, we propose a Lax-pair representation for the discrete Yajima-Oikawa system as well as its multicomponent generalization also considered in arXiv:1509.06996 and prove that it has an infinite number of conservation laws. We also derive the next higher flow of the discrete Yajima-Oikawa hierarchy, which generalizes a modified version of the Volterra lattice. Relations to two integrable discrete nonlinear Schrodinger hierarchies, the Ablowitz-Ladik hierarchy and the Konopelchenko-Chudnovsky hierarchy, are clarified.
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TL;DR: In this article, a review of the theoretical and experimental results obtained on simple magnetic model systems on magnetic lattices of dimensionality 1, 2, and 3 is presented, with particular attention paid to the approximation of these model systems in real crystals, viz how they can be realized or be expected to exist in nature.
Abstract: “…. For the truth of the conclusions of physical science, observation is the supreme Court of Appeal….” (Sir Arthur Eddington, The Philosophy of Physical Science.) In this paper we shall review the theoretical and experimental results obtained on simple magnetic model systems. We shall consider the Heisenberg, XY and Ising type of interaction (ferro and antiferromagnetic), on magnetic lattices of dimensionality 1, 2 and 3. Particular attention will be paid to the approximation of these model systems in real crystals, viz. how they can be realized or be expected to exist in nature. A large number of magnetic compounds which, according to the available experimental information, meet the requirements set by one or the other of the various models are considered and their properties discussed. Many examples will be given that demonstrate to what extent experiments on simple magnetic systems support theoretical descriptions of magnetic ordering phenomena and contribute to their understanding. It will a...
1,570 citations
TL;DR: In this article, the authors studied thermodynamic and some dynamic properties of a one-dimensional model system whose displacement field Hamiltonian is strongly anharmonic, and is representative of those used to study displacive phase transitions.
Abstract: We have studied thermodynamic and some dynamic properties of a one-dimensional-model system whose displacement field Hamiltonian is strongly anharmonic, and is representative of those used to study displacive phase transitions. By studying the classical equations of motion, we find important solutions (domain walls) which cannot be represented effectively by the usual phonon perturbation expansions. The thermodynamic properties of this system can be calculated exactly by functional integral methods. No Hartree or decoupling approximations are made nor is a temperature dependence of the Hamiltonian introduced artificially. At low temperature, the thermodynamic behavior agrees with that found from a phenomenological model in which both phonons and domain walls are included as elementary excitations. We then show that equal-time correlation functions calculated by both functional-integral and phenomenological methods agree, and that the dynamic correlation functions (calculated only phenomenologically) exhibit a spectrum with both phonon peaks and a central peak due to domain-wall motion.
685 citations
TL;DR: A survey of the properties of soliton-type solutions to non-linear wave equations appearing in various fields of physics is given in this paper, where the results of computer experiments on the dynamics of the formation and interaction (in one-space-dimensional geometry) of solit-type objects are presented at length.
499 citations
300 citations