Integrable Reductions of Manycomponent Magnetic Systems in (1,1) Dimensions

Particle-Like Excitations in Many Component Magnon-Phonon Systems

Abstract: It is shown that many sublattice isotropic XY chain with magnon-phonon interactions at the long-wave approximations may be described by generalized Zakharov's system with U(p, q) isogroup. Two-sublattice XY chain through the Jordan-Wigner transformations reduced to Su-Schriffer-Heeger coupled electron-phonon model in the quasi-one-dimensional conductor polyacetylene (CH)x theory. The Hamiltonian structure of U(p, q) Zakharov's system and its "ultrarelativistic" limit (i.e., the Yajima-Oikawa system with U(p, q) isogroup) are studied. The linear problem and the generating functionals for infinite series of additional integrals of motion for U(p, q) Yajima-Oikawa system are constructed. Four types of soliton solutions under different boundary conditions and appropriate integrals of particles number and energy are found. The quasistationary and ultra-relativistic limits are discussed.

Generalized Coherent States and the Continuous Heisenberg XYZ Model With One-Ion Anisotropy

Abstract: On the basis of the method of generalized coherent states (GCS) of the SU(2) group we formulate the reduction procedure of the quantum lattice Heisenberg XYZ model to the continuum Landau-Lifshitz model. The choice of the GCS representation is determined (i) by their proximity to the corresponding classical states and (ii) by similarity of the geometric structure of the homogeneous spaces on which the SU(2) GCS and the vector of the magnetic moment of the corresponding Landau-Lifshitz model are defined. The present reduction procedure is allowed only in the case of weak anisotropy. Accounting for a weak one-ion anisotropy leads to the Landau-Lifshitz model with a renormalized anisotropy tensor. The contribution of such an anisotropy vanishes for the s = 1/2 case.

A noncompact spin model in (2+1) dimensions

Abstract: A modified version of the Ishimori model endowed with SU(1,1) symmetry is proposed. Some classes of both rational and biperiodic exact singular solutions are obtained. These solutions are characterized by a topological charge.

New types of two component NLS-type equations

Abstract: We study MNLS related to the DIII-type symmetric spaces Applying to them Mikhailov reduction groups of the type $\mathbb{Z}_r\times \mathbb{Z}_2$ we derive new types of 2-component NLS equations These are {\bf not} counterexamples to the Zakharov-Schulman theorem because the corresponding interaction Hamiltonians depend not only on $|q_k|^2$, but also on $q_1q_2^* +q_1^* q_2$

Soliton dynamics and elastic collisions in a spin chain with an external time-dependent magnetic field

Abstract: In this paper, we investigate the nonlinear dynamics of a Heisenberg spin chain with an external time-oscillating magnetic field. Such dynamics can be described by a Landau–Lifshitz-type equation. We apply the Darboux transformation method to the linear eigenvalue problem associated with this equation, and obtain the multi-soliton solution with a purely algebraic iterative procedure. Through the analytical analysis and graphical illustrations for the solutions obtained, we discuss in detail the effects of an external magnetic field on the nonlinear wave. Under the action of an external field, although the amplitude, width and depth of soliton vary periodically with time and its symmetry property is changeable, the soliton can also propagate stably and it possesses particle-like behavior.

Experiments on simple magnetic model systems

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...

Dynamics and statistical mechanics of a one-dimensional model Hamiltonian for structural 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.

Dynamics of classical solitons (in non-integrable systems)

Abstract: A survey of the properties of soliton type solutions to non-linear wave equations appearing in various fields of physics is given. For the most part the description deals with non-integrable systems and remains at the classical level. Some general features associated with solitons-stationary states, bound states of solitons, stability, the Fermi-Pasta-Ulam problem, and so on are discussed for both one- and more-dimensional worlds. The results of computer experiments on the dynamics of the formation and interaction (in one-space-dimensional geometry) of soliton-type objects are presented at length. As an application of the general theory we speculate on solitons in plasmas and particle physics. Finally, new, spherically symmetric, oscillating solutions (“pulsons”) are examined in the framework of ϕ4 and sine-Gordon theories.