Systems Of Mathematical Physics Research Articles

Non-Lie symmetry groups of (2+1)-dimensional nonlinear systems obtained from a simple direct method

A new direct method is developed to find finite symmetry groups of nonlinear mathematical physics systems. Using the direct method for the well-known (2+1)-dimensional Kadomtsev–Petviashvili equation and the Ablowitz–Kaup–Newell–Segur system, both the Lie point symmetry groups and the non-Lie symmetry groups are obtained. The Lie symmetry groups obtained via traditional Lie approaches are only special cases. Furthermore, the expressions of the exact finite transformations of the Lie groups are much simpler than those obtained via the standard approaches.

Quaternionic factorization of the Schrödinger operator and its applications to some first-order systems of mathematical physics

We consider the following first order systems of mathematical physics. 1.The Dirac equation with scalar potential. 2.The Dirac equation with electric potential. 3.The Dirac equation with pseudoscalar potential. 4.The system describing non-linear force free magnetic fields or Beltrami fields with nonconstant proportionality factor. 5.The Maxwell equations for slowly changing media. 6.The static Maxwell system. We show that all this variety of first order systems reduces to a single quaternionic equation the analysis of which in its turn reduces to the solution of a Schroedinger equation with biquaternionic potential. In some important situations the biquaternionic potential can be diagonalized and converted into scalar potentials.

Geometrical interpretation of Benenti systems

We show that the following two separately developed theories, the theory of Benenti systems in mathematical physics and the theory of projectively equivalent metrics in classical differential geometry, study essentially the same object. Combining methods and results from these two theories, one can prove the commutative integrability of projectively equivalent pseudo-Riemannian metrics and construct infinitely many new Hamiltonian systems, integrable in the classical and in the quantum sense.

Averaging of trajectory attractors of evolution equations with rapidly oscillating terms

Evolution equations containing rapidly oscillating terms with respect to the spatial variables or the time variable are considered. The trajectory attractors of these equations are proved to approach the trajectory attractors of the equations whose terms are the averages of the corresponding terms of the original equations. The corresponding Cauchy problems are not assumed here to be uniquely soluble. At the same time if the Cauchy problems for the equations under consideration are uniquely soluble, then they generate semigroups having global attractors. These global attractors also converge to the global attractors of the averaged equations in the corresponding spaces. These results are applied to the following equations and systems of mathematical physics: the 3D and 2D Navier-Stokes systems with rapidly oscillating external forces, reaction-diffusion systems, the complex Ginzburg-Landau equation, the generalized Chafee-Infante equation, and dissipative hyperbolic equations with rapidly oscillating terms and coefficients.

Reduction and a Concentration-Compactness Principle for Energy-Casimir Functionals

Energy-Casimir functionals are a useful tool for the construction of steady states and the analysis of their nonlinear stability properties for a variety of conservative systems in mathematical physics. Recently, Y. Guo and the author employed them to construct stable steady states for the Vlasov--Poisson system in stellar dynamics, where the energy-Casimir functionals act on number density functions on phase space. In the present paper we construct natural, reduced functionals which act on mass densities on space and study compactness properties and the existence of minimizers in this context. This puts the techniques developed by Y. Guo and the author into a more general framework. We recover the concentration-compactness principle due to P.-L. Lions [Ann. Inst. H. Poincaré Anal. Non Linéaire, 1 (1984), pp. 109--145] in a more specific setting and connect our stability analysis with that of G. Wolansky [Ann. Inst. H. Poincaré Anal. Non Linéaire, 16 (1999), pp. 15--48].

Bifurcations and selection of equilibria in a simple cosymmetric model of filtrational convection

A three-dimensional set of ordinary differential equations that constitutes a simple abstract model of Darcy convection is investigated. The model reproduces a number of effects that are typical for dynamic systems with nontrivial cosymmetry. Nontrivial cosymmetry can give rise to a continuous family of equilibria where, in this case, the equilibrium stability spectrum varies along the family. The family of equilibria and its stability are examined analytically, and special bifurcations that occur in the system are investigated. It is shown that discrete and continual symmetries, called "flash symmetries," can be present in the system for certain parameter values. Computer experiments on the selection of equilibria in the symmetric and cosymmetric cases have been carried out. They showed that, for initial points that are far enough from a cycle of equilibria, the neighborhood of a single equilibrium is established in the case of cosymmetry, but all the equilibria are equivalent in the case of symmetry. The authors hope that these results, as well as the formulation of the problems and the approach to their solution, will serve as a sample in the investigation of more complex systems in mathematical physics. (c) 1999 American Institute of Physics.

On Uniqueness in a Lateral Cauchy Problem with Multiple Characteristics

where f ({w)=1 2{w {w. Indeed, we are able to derive from our results on scalar operators in this paper and in the paper [4] that a regular (C is enough) solution to the system (0.2) is uniquely determined in the domain Qo by the Cauchy data on 1 provided f # C. The domain Qo /Q depends on 1 and is explicitly described by some quadratic function. We expect that other important systems of mathematical physics can be handled in a similar manner. As shown in [6, 7] uniqueness in the Cauchy article no. DE963227

An Integrated Approach to Dynamic Analysis of the Ring Spinning Process

This paper will show that the theory of ring spinning developed by Batra et al. and subsequently by Fraser can be used to explain recent experimental results obtained at the SRRC. In particular, Fraser showed that the quasi-stationary, nonlinear equations of motion relevant to ring spinning, including the effect of centripetal acceleration and air drag force, developed earlier by several investigators exhibit a bifurcation phe nomenon typical of many other nonlinear systems in mathematical physics. This investigation shows that the bifurcation analysis applied in a way that simulates for mation of the bobbin, even a chase of the bobbin, reveals meta-stability in parametric space, which can be used to explain the instabilities in free (no control rings) balloon profiles observed experimentally.

Adiabatic invariants of nonlinear dynamic systems with small parameter

We develop a symplectic method of finding the adiabatic invariants of nonlinear dynamic systems with small parameter. We show that a necessary and sufficient condition for the existence of quasi-Hamiltonian adiabatic invariants of nonlinear dynamic systems with regular dependence on a small parameter is that the Cauchy problem be well-posed for an equation of Lax type in the class of nongradient local functionals on the cotangent manifold of the phase space. It is established that scalar nonlinear dynamic systems always have a priori complete evolution invariants, not only adiabatic invariants. We also consider typical applications in hydrodynamics and oscillatory systems of mathematical physics.

Integrated semigroups and the cauchy problem for systems in Lp spaces

In this note we prove that (under suitable hypotheses) every homogeneous differential operator on L p( R n) N , corresponding to a system which is well-posed in L 2( R n) N , generates an α-times integrated semigroup on L p( R n) N (1 < p < ∞) whenever α > n ¦ 1 2 − 1 p ¦ . For some special systems of mathematical physics, such as the wave equation or Maxwell's equations this constant can be improved to be (n − 1) ¦ 1 2 − 1 p ¦ .

Algebraic scheme of discrete approximations of linear and nonlinear dynamical systems of mathematical physics

Algebraic scheme of discrete approximations of linear and nonlinear dynamical systems of mathematical physics

Bilinear dynamical systems in mathematical physics

Bilinear dynamical systems in mathematical physics

Time evolution and fluctuations of the probability density and entropy function for a class of nonlinear stochastic systems in mathematical physics

A mathematical theory is proposed to describe the time evolution and fluctuations for the probability density and the entropy functions related to the state variable of nonlinear stochastic systems. The class of physical systems whose evolution can be modelled by nonlinear stochastic ordinary differential equations is considered and the evolution equation for the above specified functions is derived under suitable regularity conditions. Quantitative results in the applications are supplied on the basis of an analysis using the Adomian's decomposition method.

Investigation and applications of pansystems recognition theory and pansystems-operations research of large-scale systems (I)

In this work, a sort of recognition theory and operations research of large scale systems are developed within the framework of pansystems methodology. We establish a series of theorems concerning pansystems relations, which discuss some fundamental problems of interdisciplines from the viewpoint of generalized system-transformation-symmetry in things mechanism, and are connected closely with mathematical physics systems thinking science bioecological sciences and the methodological investigation of mechanical foundation. This paper offers 100 pansystems theorems.

The Method of Arbitrary Functions

The method of arbitrary functions is a mathematical technique for the determination of probability distributions. It has been used to justify a unique and objectively interpreted concept of probability in games of chance of a mechanical type, and in the theory of dynamical systems of mathematical physics. Relatively recent results in the ergodic theory of classical dynamical systems will be useful for a re-evaluation of the method of arbitrary or random functions. Towards the end of this paper, we will try to show how, and to what degree, these new results justify the hopes of the pioneers of the method. The literature on probability in dynamical systems, which goes back almost one hundred years, has been unjustly forgotten. It forms an important chapter in the foundations of probability, and includes technical and interpretative insights of great relevance. It may be surprising to find out that probabilities in games of chance which are supposed to be mechanical systems are given an objective interpretation. Nowadays many a philosopher of probability advocates the view that a theory of genuine chance phenomena has to be based on a quantum-mechanically motivated notion of indeterminism. The possibility of 'a causal theory of probability' shows that this propensity point of view, whenever it is applied on a macroscopic level, can be rivalled by a view in which a non-classical notion of indeterminism is not needed. The literature on the method of arbitrary functions and ergodic theory contains studies of specific systems with given equations of motion. Beginning with the work of Ya. Sinai in 1963, the existence of dynamical systems which are measure-theoretically isomorphic to certain stochastic processes has been established. These rigorous results of mathematical physics, and their relevance for the interpretation of probability, have so far been left largely unnoticed by philosophers. Now

Variational Representations in Reactor Physics Derived from a Physical Principle

A physical axiom is advanced that relates density of neutrons and their individual contribution to operationally determinable behavior of a reactor. The variational principle derived from this axiom is of a general form applicable to systems in which time dependency of coefficients of equations prevents a separation into conventional eigenfunctions and eigenvalues. The physical significance of independent variation of two field functions is investigated. The treatment of nonseparable systems and variational principle to which we are led are both independent of any particular physical model employed to represent system and appear to be applicable to a variety of nonconservative, continuous, and time-dependent systems in mathematical physics. The more well-known properties of separable problem are derived from principle as the exception proving rule in an attempt to associate physical meaning with commnonly . employed forms. Thus a discussion is given of relation of Greens function to both fields and Joint Error is introduced as a criterion for completeness of biorthegonal sets. Although variational principle derived is not applicable to variation of coefficients of equations through nonlinearities, it is indicated how themore » present approach may be extended to account for nonlinearities. (auth)« less