Nonlinear chiral models: Soliton solutions and spatio-temporal chaos

Nonlinear effective Lagrangian models with a chiral symmetry have been used to describe strong interactions at low energy, for a long time. The Skyrme model and the chiral quark-meson model are two such models, which have soliton solutions which can be identified with the baryons. We describe the various kinds of soliton states in these nonlinear models and discuss their physical significance and uses in this review. We also study these models from the view point of classical nonlinar dynamical systems. We consider fluctuations around theB=1 soliton solutions of these models (B, being the baryon number) and solve the spherically symmetric, time-dependent systems. Numerical studies indicate that the phase space around the Skyrme soliton solution exhibits spatio-temporal chaos. It is remarkable that topological solitons signifying stability/order and spatio-temporal chaos coexist in this model. In contrast with this, the soliton of the quark-meson model is stable even for large perturbations.     

Analytic instanton solutions in two-dimensional field models

Two-dimensional scalar field models with nontrivial potential determined in bounded spatial regions are considered. Exact analytic instanton solutions for these models and soliton solutions for similar three-dimensional systems are obtained.     

Absence of static solutions in source-free Yang-Mills theory

The vanishing of self-stress for static systems excludes finite energy time-independent solutions of source-free Yang-Mills models (with compact gauge groups), except in five dimensions. Static solutions of Abelian self-interacting models of the Born-Infeld type are also forbidden.     

Generalized variational principles,global weak solutions and behavior with random initial data for systems of conservation laws arising in adhesion particle dynamics

We study systems of conservation laws arising in two models of adhesion particle dynamics. The first is the system of free particles which stick under collision. The second is a system of gravitationally interacting particles which also stick under collision. In both cases, mass and momentum are conserved at the collisions, so the dynamics is described by 2×2 systems of conservations laws. We show that for these systems, global weak solutions can be constructed explicitly using the initial data by a procedure analogous to the Lax-Oleinik variational principle for scalar conservation laws. However, this weak solution is not unique among weak solutions satisfying the standard entropy condition. We also study a modified gravitational model in which, instead of momentum, some other weighted velocity is conserved at collisions. For this model, we prove both existence and uniqueness of global weak solutions. We then study the qualitative behavior of the solutions with random initial data. We show that for continuous but nowhere differentiable random initial velocities, all masses immediately concentrate on points even though they were continuously distributed initially, and the set of shock locations is dense.     

Vibration analysis of a wheel composed of a ring and a wheel-plate modelled as a three-parameter elastic foundation

The free in-plane vibrations of circular rings with wheel-plates as generalised elastic foundations are studied using analytical methods and numerical simulations. The three-parameter Winkler elastic layer is proposed as a mathematical model of the foundation. The effects of rotary inertia and shear deformation are included in the analytical model of the system. The motion equations of systems are derived on the basis of the thin ring theory and Timoshenko?s theory. The separation of variables method is used to find general solutions to the free vibrations. Elaborated analytical models are used to determine the natural frequencies and the natural mode shapes of vibrations of an arbitrarily chosen set of simplified models of aviation gears and railway wheels. The eigenvalue problem is formulated and solved by using a finite element representation for each simplified model. The results for these models are discussed and compared. The proposed solutions are verified by experimental investigation. It is important to note that the solutions proposed here could be useful to engineers dealing with the dynamics of aviation gears, railway wheels and other circular ring systems.     

Exactly solvable models: The road towards a rigorous treatment of phase transitions in finite systems

The exact analytical solutions of a variety of statistical models recently obtained for finite systems are thoroughly discussed. Among them are a constrained version of the statistical multifragmentation model, the Gas of Bags Model, and the Hills and Dales Model of surface partition. The finite volume analytical solutions of these models were obtained by a novel powerful mathematical method, the Laplace-Fourier transform. Thus, the Laplace-Fourier transform allows one to study the nuclear matter equation of state, the equation of state of hadronic matter and quark gluon plasma, and the surface entropy of large clusters on the same footing. A complete analysis of the isobaric partition singularities of these models is done for finite systems. The developed formalism allows us, for the first time, to exactly define the finite volume analogs of gaseous, liquid, and mixed phases of these models from the first principles of statistical mechanics, and to demonstrate the pitfalls of earlier works. The found solutions may be used for building up a new theoretical apparatus to rigorously study phase transitions in finite systems. The strategic directions of future research opened by these exact results are also discussed. The text was submitted by the author in English.     

Mechanical systems with Poincaré invariance

Some years ago Ruijsenaars and Schneider initiated the study of mechanical systems exhibiting an action of the Poincaré algebra. The systems they discovered were far richer: their models were actually integrable and possessed a natural quantum version. We follow this early work finding and classifying mechanical systems with such an action. New solutions are found together with a new class of models exhibiting an action of the Galilean algebra. These are related to the functional identities underlying the various Hirzebruch genera. The quantum mechanics is also discussed.     

On the nonlinear instability of the solutions of hydrodynamic-type systems

New exact solutions (including periodic) of three-dimensional nonstationary Navier-Stokes equations containing arbitrary functions are described. The problems of the nonlinear stability/instability of the solutions have been analyzed. It has been found that a feature of a wide class of the solutions of hydrodynamic-type systems is their instability. It has been shown that instability can occur not only at sufficiently large Reynolds numbers, but also at arbitrary small Reynolds numbers (and can be independent of the fluid velocity profile). A general physical interpretation of the solution under consideration is given. It is important to note that the instability of the solutions has been proven using a new exact method (without any assumptions and approximations), which can be useful for analyzing other nonlinear physical models and phenomena.     

Intraband discrete breathers in disordered nonlinear systems. I. Delocalization

The existence of Discrete Breathers or DBs (also called Intrinsic Localized Modes or ILMs) and multibreathers, is investigated in a simple one-dimensional chain of random anharmonic oscillators with quartic potentials coupled by springs. When the breather frequency is outside and above the linearized (phonon) spectrum, the existence theorems and numerical methods previously used in periodic nonlinear models for finding time-periodic and spatially localized solutions, hold identically in random nonlinear systems. These solutions are extraband discrete breathers (EDBs). When the frequencies penetrate inside the linearized spectrum, the existence theorems do not hold. Our numerical investigations demonstrate that the strict continuation of (localized) EDBs as intraband discrete breathers (IDBs) is impossible because of cascades of bifurcations generating many discontinuities. A detailed analysis of these bifurcations for small systems with increasing sizes, shows that only a relatively small subset of the spatially extended multibreathers can be strictly continued while their frequency varies inside the phonon spectrum. We propose an ansatz for finding the coding sequences of these solutions and continuing safely these multibreathers in finite systems of any size. This continuation ends at a lower limit frequency where the solution annihilates through a bifurcation with another multibreather. A smaller subset of these multibreather solutions can be continued to amplitude zero and become linear localized modes at this limit. Conversely, any linear localized mode can be continued when increasing its frequency as an extended multibreather. Extrapolation of these results to infinite systems yields the main conclusion of this first part which is that nonlinearity in disordered systems (with localized eigenmodes only) restores their capability of energy transportation by generating infinitely many spatially extended time-periodic solutions. This approach yields mainly spatially extended solutions, except sometimes at their bifurcation points. In the second part of this work, which is presented in our next article, we develop an accurate method for calculating in situ localized IDBs.     

Zero temperature solutions of the Edwards-Anderson model in random Husimi lattices

We solve the Edwards-Anderson model (EA) in different Husimi lattices using the cavity method at replica symmetric (RS) and 1-step of replica symmetry breaking (1RSB) levels. We show that, at T = 0, the structure of the solution space depends on the parity of the loop sizes. Husimi lattices with odd loop sizes may have a trivial paramagnetic solution thermodynamically relevant for highly frustrated systems while, in Husimi lattices with even loop sizes, this solution is absent. The range of stability under 1RSB perturbations of this and other RS solutions is computed analytically (when possible) or numerically. We also study the transition from 1RSB solutions to paramagnetic and ferromagnetic RS solutions. Finally we compare the solutions of the EA model in Husimi lattices with that on the (short loops free) Bethe lattices, showing that already for loop sizes of order 8 both models behave similarly.     

Rogue waves in nonlinear Schrödinger models with variable coefficients: Application to Bose–Einstein condensates

We explore the form of rogue wave solutions in a select set of case examples of nonlinear Schrödinger equations with variable coefficients. We focus on systems with constant dispersion, and present three different models that describe atomic Bose–Einstein condensates in different experimentally relevant settings. For these models, we identify exact rogue wave solutions. Our analytical findings are corroborated by direct numerical integration of the original equations, performed by two different schemes. Very good agreement between numerical results and analytical predictions for the emergence of the rogue waves is identified. Additionally, the nontrivial fate of small numerically induced perturbations to the exact rogue wave solutions is also discussed.     

Field-theoretic extended particles in two space dimensions

A family of Lagrangian models of two scalar fields in (2 + 1) dimensions is studied. All their localized static classical solutions are obtained, and interpreted as representing systems of extended particles. The models are generalized by taking into account general-relativistic gravirational coupling, for which case the general static solutions are also obtained explicitly.     

Cosmological Models and Centre Manifold Theory

Centre manifold theory is applied to some dynamical systems arising from spatially homogeneous cosmological models. Detailed information is obtained concerning the late-time behaviour of solutions of the Einstein equations of Bianchi type III with collisionless matter. In addition some statements in the literature on solutions of the Einstein equations coupled to a massive scalar field are proved rigorously.     

Quasilinear hyperbolic systems

We construct global solutions for quasilinear hyperbolic systems and study their asymptotic behaviors. The systems include models of gas flows in a variable area duct and flows with a moving source. Our analysis is based on a numerical scheme which generalizes the Glimm scheme for hyperbolic conservation laws.Partially supported by National Science Foundation Grant NSF MCS78-2202     

General properties of geophysical hydrodynamics models

The influence of simplifying assumptions used in geophysical hydrodynamics models on the general properties of space and time, as well as on the fundamental physical principles that underlie the mechanical systems, is studied. Analysis of the general properties of such models is supplemented by investigation of the peculiarities introduced by stratification into the solutions of nonlinear stationary models. The possibilities for representing the solutions as finite and infinite Fourier series in cylindrical coordinates are explored.     

Free transverse vibration analysis of an elastically connected annular and circular double-membrane compound system

In this paper, free transversal vibrations of a systems of two annular and circular membranes connected by a Winkler elastic layer are studied using analytical methods and numerical simulation. At first the motion of each system is described by two homogeneous partial differential equations. The general solutions of the free vibrations are derived by the Bernoulli-Fourier method and the boundary problems are solved. The natural frequencies and natural mode shapes of vibrations of systems under consideration are determined. The investigation of free vibrations prove that the double-membrane systems execute two kinds of vibrations, synchronous and asynchronous. Then for each system two models formulated by using finite element representations are prepared. The FE models are manually tuned to reduce the difference between the natural frequencies of the analytical solutions and the natural frequencies of the FE model calculations, respectively.     

Independent particle model of spontaneous symmetry breaking in planar π-electron systems

The singlet stability of symmetry adapted (SA), restricted Hartree-Fock (RHF) solutions, and the implied symmetry breaking for several planar, π-electron systems, is investigated using the semiempirical Pariser-Parr-Pople Hamiltonian in the whole range of the coupling constant. We focus here on highly symmetric cyclic polyenes C₁₀H₁₀ and C₁₄H₁₄ and their various distorted analogues of lower symmetry, in particular on the perimeter models of naphthalene and anthracene (p-naphthalene and p-anthracene) modeling the so-called [n]-annulenes. Relying on earlier results for general systems with conjugated double-bonds, we explore the character and properties of both the SA and broken-symmetry (BS) RHF solutions for these systems and relate their behavior to those of highly symmetric cyclic polyenes and corresponding polyacenes. In this way we are able to provide a better understanding of the spontaneous symmetry breaking in these systems at the Hartree-Fock level of approximation.     

Exact Solutions of the Coupled KdV system via a Formally Variable Separation Approach

Most of the nonlinear physics systems are essentially nonintegrable.There in no very doog analytical approach to solve nonintegrable system.The variable separation approach is a powerful method in linear physics.In this letter,the formal variable separation approach is established to solve the generalized nonlinear mathematical physics equation.The method is valid not only for integrable models but also for nonintegrable models.Taking a nonintegrable coupled KdV equation system as a simple example,abundant solitary wave solutions and conoid wave solutions are revealed.     

Nonisothermal absorption in thermotransformers

The results obtained using simple models of combined heat and mass transfer in nonisothermal vapor or gas absorption by aqueous solutions are generalized. The models take into account specific properties of two-phase binary systems used in absorbers of thermotransformers (heat pumps and refrigerators). Some authors have already cited certain of the results, including criterial dependences obtained via exact solutions, in reference books and monographs. This paper represents a wider spectrum of problems. The Russian, more detailed version of the article, was published as a book.     

Oscillatory behaviour of closed isotropic models in second order gravity theory

Homogeneous and isotropic models are studied in the Jordan frame of the second order gravity theory. The late time evolution of the models is analysed with the methods of the dynamical systems. The normal form of the dynamical system has periodic solutions for a large set of initial conditions. This implies that an initially expanding closed isotropic universe may exhibit oscillatory behaviour.