The dynamics of spatiotemporal modulations

The modulational instability of traveling waves is often thought to be a crucial point in the mechanism of transition to space-time disorder and turbulence. The aim of this paper is to study the effect of spatiotemporal modulations on some dynamics u(0)(x,t), which may occur as an instability process when a control parameter varies, for instance. We analyze the properties of the modulated dynamics of the form g(1)(x)g(2)(t)u(0)(x,t) compared to those of the reference dynamics u(0)(x,t), using operator theory. We show that, if the reference dynamics is invariant under some space-time symmetry in the sense of Ref. [J. Nonlinear Sci. 2, 183 (1992)], the modulation has the effect of either deforming this symmetry or breaking it, depending on whether the corresponding operator remains unitary or not. We also demonstrate that the smallest Euclidean space containing the modulated dynamics has a dimension smaller than or equal to the smallest Euclidean space containing u(0)(x,t). The previous results are then applied to the case of modulated uniformly traveling waves. While the spatiotemporal translation invariance of the wave never persists in the presence of a modulation, the existence of a spatiotemporal symmetry depends on the resonance of the Fourier sidebands due to the modulation. In case of nonresonance, a spatiotemporal symmetry exists and is explicitly determined. In this situation, the modulated wave and the carrier wave have the same spectrum (up to a normalization factor), the same entropy, and the spatial (resp., temporal) two-point correlation is deformed only by the spatial (resp., temporal) modulation. (c) 1995 American Institute of Physics.     

Time-Dependent Lagrangians Invariant by a Vector Field

We study the reduction of nonautonomous regular Lagrangian systems by symmetries, which are generated by vector fields associated with connections in the configuration bundle of the system Q × . These kind of symmetries generalize the idea of time-invariance (which corresponds to taking the trivial connection in the above trivial bundle).     

Nonclassical Symmetries for a Class of Reaction-Diffusion Equations: the Method of Heir-Equations

The nonclassical symmetries method is applied to a class of reaction-diffusion equations with nonlinear source, i.e. u _t =u _xx +cu _x +R(u, x). Several cases are obtained by using suitable solutions of the heir-equations as described in [M.C. Nucci, Nonclassical symmetries as special solutions of heir-equations, J. Math. Anal. Appl. 279 (2003) 168–179].     

Pseudo-Hermitian Hamiltonians and Generalized Involutory Symmetries

We propose definitions of generalized parity (P), time-reversal (T) and charge-conjugation (C) operators such, that any diagonalizable pseudo-Hermitian Hamiltonian is invariant under the involutory symmetries C, TP, and CPT. We inquire about the peculiarities of such symmetries showing that these constitute the P-unitary and P-antiunitary symmetry generators. Moreover, we give a necessary and sufficient condition for diagonalizable pseudo-Hermitian Hamiltonians to admit P-pseudounitary and P-pseudoantiunitary symmetries.     

Stability of proton and maximally symmetric minimal unification model for basic forces and building blocks of matter

With the hypothesis that all independent degrees of freedom of basic building blocks should be treated equally on the same footing and correlated by a possible maximal symmetry, we arrive at a 4-dimensional space-time unification model. In this model the basic building blocks are Majorana fermions in the spinor repre- sentation of 14-dimensional quantum space-time with a gauge symmetry GM4D = SO(1,3)×SU(32)×U(1)A×SU(3)F. The model leads to new physics including mirror particles of the standard model. It enables us to issue some fundamental questions that include: why our living space-time is 4-dimensional, why parity is not con- served in our world, how the stability of proton is, what the origin of CP violation is and what the dark matter can be.     

The CPT Group of the Dirac Field

Using the standard representation of the Dirac equation, we show that, up to signs, there exist only two sets of consistent solutions for the matrices of charge conjugation (C), parity (P), and time reversal (T), which give the transformation of fields , and , where and . These sets are given by , , and , , . Then , and two successive applications of the parity transformation to fermion fields necessarily amount to a 2 rotation. Each of these sets generates a non abelian group of 16 elements, respectively, and , which are non isomorphic subgroups of the Dirac algebra, which, being a Clifford algebra, gives a geometric nature to the generators, in particular to charge conjugation. It turns out that and , where is the dihedral group of eight elements, the group of symmetries of the square, and 16E is a non trivial extension of by , isomorphic to a semidirect product of these groups; S₆ and S₈ are the symmetric groups of six and eight elements. The matrices are also given in the Weyl representation, suitable for taking the massless limit, and in the Majorana representation, describing self-conjugate fields. Instead, the quantum operators C, P and T, acting on the Hilbert space, generate a unique group , which we call the CPT group of the Dirac field. This group, however, is compatible only with the second of the above two matrix solutions, namely with , which is then called the matrix CPT group. It turns out that , where is the dicyclic group of 8 elements and S₁₀ is the symmetric group of 10 elements. Since , the quaternion group, and , the 0-sphere, then .     

Statistical symmetries and its impact on new decay modes and integral invariants of decaying turbulence

Classical decay laws of isotropic turbulence usually derived from the von Kármán–Howarth equation are essentially based on two paradigms. First, scaling symmetries of space and time, both tracing back to the Navier–Stokes equations in the limit of large Reynolds numbers (or r?η), give rise to a temporal power-law decay for the turbulent kinetic energy and at the same time an algebraic growth of the integral length scale at an exponent that is uniquely coupled to the latter energy decay. Second, global invariants such as Birkhoff or Loitsianskii integrals determine the exponent of both power laws. We presently show that this class of decay laws may be considerably extended considering the entire set of multi-point correlation equations that admit a much wider class of symmetries. It was recently shown that these new symmetries are of paramount importance, e.g. in deriving the logarithmic law of the wall being an analytic solution of the multi-point equations. For the present case, it is particularly an additional scaling group, which we call statistical scaling group, that gives rise to two additional families of ‘canonical’ decay laws including those with an exponential characteristic for both the kinetic energy and the integral length scale. Finally, a second rather generic group admitted by all linear differential equations corresponding to the superposition principle induces an infinite set of scaling laws of rather complex form that may match rather generic initial conditions. All scaling laws are analyzed in the light of the above-mentioned integral invariants that have been further extended in the present contribution to an exponential-type invariant.     

Symmetries and conservation laws of the damped harmonic oscillator

We work with a formulation of Noether-symmetry analysis which uses the properties of infinitesimal point transformations in the space-time variables to establish the association between symmetries and conservation laws of a dynamical system. Here symmetries are expressed in the form of generators. We have studied the variational or Noether symmetries of the damped harmonic oscillator representing it by an explicitly time-dependent Lagrangian and found that a five-parameter group of transformations leaves the action integral invariant. Amongst the associated conserved quantities only two are found to be functionally independent. These two conserved quantities determine the solution of the problem and correspond to a two-parameter Abelian subgroup.      

Quantization of space-time and the corresponding quantum mechanics

An axiomatic framework for describing general space-time models is presented. Space-time models to which irreducible propositional systems belong as causal logics are quantum (q) theoretically interpretable and their event spaces are Hilbert spaces. Such aq space-time is proposed via a canonical quantization. As a basic assumption, the time t and the radial coordinate r of aq particle satisfy the canonical commutation relation [t,r]=±i . The two cases will be considered simultaneously. In that case the event space is the Hilbert space L²(³). Unitary symmetries consist of Poincaré-like symmetries (translations, rotations, and inversion) and of gauge-like symmetries. Space inversion implies time inversion. Thisq space-time reveals a confinement phenomenon: Theq particle is confined in an size region of Minkowski space at any time. One particle mechanics overq space-time provides mass eigenvalue equations for elementary particles. Prugoveki's stochasticq mechanics andq space-time offer a natural way for introducing and interpreting consistently such aq space-time andq particles existing in it. The mass eigenstates ofq particles generate Prugoveki's extended elementary particles. When 0, these particles shrink to point particles and is recovered as the classical (c) limit ofq space-time. Conceptual considerations favor the case [t,r]=+i , and applications in hadron physics give the fit 2/5 fermi/GeV.This paper is a revised version of the author's work, Quantization of Space-time and the Corresponding Quantum Mechanics (Part I), report KFKI-1981-48.     

Recursion operators and bi-Hamiltonian structures in multidimensions. II

We analyze further the algebraic properties of bi-Hamiltonian systems in two spatial and one temporal dimensions. By utilizing the Lie algebra of certain basic (starting) symmetry operators we show that these equations possess infinitely many time dependent symmetries and constants of motion. The master symmetries for these equations are simply derived within our formalism. Furthermore, certain new functionsT ₁₂ are introduced, which algorithmically imply recursion operators ₁₂. Finally the theory presented here and in a previous paper is both motivated and verified by regarding multidimensional equations as certain singular limits of equations in one spatial dimension.     

On sufficient conditions of locality for hierarchies of symmetries of evolution systems

We present sufficient conditions ensuring the locality of hierarchies of symmetries generated by repeated commutation of master symmetry with a seed symmetry. These conditions are applicable to a large class of (1+1)-dimensional evolution systems. Our results can also be used for proving that the time-independent part of a suitable linear-in-time symmetry is a nontrivial master symmetry and hence the system in question has infinitely many symmetries and is integrable.     

Noncoassociative twisting: Its application to κ-Poincaré algebra

It is shown that the use of noncoassociative coproduct allows us to simplify the structure of-Poincaré algebra — it contains nondeformedD=3 Euclidean quasi-bialgebra. We obtain by the dual construction the commuting, nonassociativeD=4 space-time.Presented at the 4th international Colloquium Quantum Groups and Integrable Systems, Prague, 22–24 June, 1995I would like to thank Profs. J. Lukierski and V.N. Tolstoy for valuable comments.     

The Thermodynamical Arrow of Time: Reinterpreting the Boltzmann–Schuetz Argument

The recent surge of interest in the origin of the temporal asymmetry of thermodynamical systems (including the accessible part of the universe itself) has put forward two possible explanatory approaches to this age-old problem. Hereby we show that there is a third possible alternative, based on the generalization of the classical (Boltzmann–Schuetz) anthropic fluctuation picture of the origin of the perceived entropy gradient. This alternative (which we dub the Acausal-Anthropic approach) is based on accepting Boltzmann's statistical measure at its face value, and accomodating it within the quantum cosmological concept of the multiverse. We argue that conventional objections raised against the Boltzmann–Schuetz view are less forceful and serious than it is usually assumed. A fortiori, they are incapable of rendering the generalized theory untenable. On the contrary, this analysis highlights some of the other advantages of the multiverse approach to the thermodynamical arrow of time.     

Construction of Sources for Majumdar-Papapetrou Spacetimes

We study Majumdar-Papapetrou solutions for the 3 + 1 Einstein-Maxwell equations, with charged dust acting as the external source for the fields. The spherically symmetric solution of Gürses is considered in detail. We introduce new parameters that simplify the construction of class C ¹, singularity-free geometries. The arising sources are bounded or unbounded, and the redshift of light signals allows an observer at spatial infinity to distinguish these cases. We find out an interesting affinity between the conformastatic metric and some homothetic and matter collineations. The associated geometric symmetries provide us with distinctive solutions that can be used to construct non-singular sources for Majumdar-Papapetrou spacetimes.     

The physical role of gravitational and gauge degrees of freedom in general relativity — I: Dynamical synchronization and generalized inertial effects

This is the first of a couple of papers in which the peculiar capabilities of the Hamiltonian approach to general relativity are exploited to get both new results concerning specific technical issues, and new insights about old foundational problems of the theory. The first paper includes: (1) a critical analysis of the various concepts of symmetry related to the Einstein-Hilbert Lagrangian viewpoint on the one hand, and to the Hamiltonian viewpoint, on the other. This analysis leads, in particular, to a re-interpretation of active diffeomorphisms as passive and metric-dependent dynamical symmetries of Einstein's equations, a re-interpretation which enables to disclose the (not widely known)) connection of a subgroup of them to Hamiltonian gauge transformations on-shell; (2) a re-visitation of the canonical reduction of the ADM formulation of general relativity, with particular emphasis on the geometro-dynamical effects of the gauge-fixing procedure, which amounts to the definition of a global non-inertial, space-time laboratory. This analysis discloses the peculiar dynamical nature that the traditional definition of distant simultaneity and clock-synchronization assume in general relativity, as well as the gauge relatedness of the “conventions” which generalize the classical Einstein's convention. (3) a clarification of the physical role of Dirac and gauge variables, as their being related to tidal-like and generalized inertial effects, respectively. This clarification is mainly due to the fact that, unlike the standard formulations of the equivalence principle, the Hamiltonian formalism allows to define a generalized notion of “force” in general relativity in a natural way.     

Symmetries and correlation inequalities for classicaln-vector models

We describe a new class of single spin measures on then-dimensional sphereS _r ⁿ of radiusr (n 4) for which Lebowitz-type [J. Lebowitz,J. Stat. Phys. 16:463 (1977)] inequalities hold. This is achieved by an appropriate parametrization ofS _r ⁿ . The above class includes the uniform measures onxs Rⁿ ¦x¦ r for any 0 p r. The second topic of this paper is an abstract formulation of the first Griffiths inequality [R. B. Griffiths,J. Math. Phys. 8:478 (1967)] and the underlying symmetry property.     

微扰Kepler系统轨道微分方程的近似Lie对称性与近似不变量

把极角θ视为独立变量,得到Kepler系统的轨道微分方程.首先讨论Kepler系统轨道微分方程的Lie对称性和不变量,微扰Kepler系统轨道微分方程的精确Lie对称性和精确不变量,其次讨论微扰Kepler系统轨道微分方程的近似Lie对称性和近似不变量,并得到了微扰Kepler系统的9个一阶近似Lie对称性和6个一阶近似不变量,其中1个实为精确不变量,而其余5个分别等于微扰系数ε乘以Kepler系统相应的5个不变量。     

Differential Equations Invariant Under Conditional Symmetries

Nonlinear PDE’s having given conditional symmetries are constructed. They are obtained starting from the invariants of the conditional symmetry generator and imposing the extra condition given by the characteristic of the symmetry. Series of examples starting from the Boussinesq and including non-autonomous Korteweg–de Vries like equations are given to show and clarify the methodology introduced.     

General relativity in a (2+1)-dimensional space-time: An electrically charged solution

We have found a static electrically charged solution to the Einstein-Maxwell equations in a (2+1)-dimensional space-time. Studies of general relativity in lower dimensional space-times provide many new insights and a simplified arena for doing quantum mechanics. In (2+1)-dimensional space-time, solutions to the vacuum field equations are locally flat (point masses are conical sigularities), but when electromagnetic fields are presentT ^ab O and the solutions are curved. For a static chargeQ we find andds ²= –(kQ ² /2)In(r _c /r)dt ² + (2/kQ ²[ln(r _c /r)]^–1 dr ² +r ² d ² wherer _c is a constant. There is a horizon atr =r _c like the inner horizon of the Reisner-Nordström solution. We have produced a Kruskal extension of this metric which shows two static regions (I and III) withr <r _c and two dynamical regions (II and IV) withr>r _c . A spacelike slice across regions I and III shows a football-shaped universe with chargeQ at one end and –Q at the other. Slices in the dynamical regions (II and IV) show a cylindrical universe that is expanding in region II and contracting in region IV. Electromagnetic solutions to the Einstein-Maxwell field equations in lower dimensional space-times can be used to provide new insights into Kaluza-Klein theories. In terms of the Kaluza-Klein theory, for example, electromagnetic radiation in a (2+1)-dimensional space-time is really gravitational radiation in the associated (3+1)-dimensional Kaluza-Klein space-time. According to Kaluza Klein theory the absence of gravitational radiation in (2+1)-dimensional space-time implies (correctly) the absence of electromagnetic radiation in (1+1)-dimensional space-time.     

Wave propagation phenomena from a spatiotemporal viewpoint: Resonances and bifurcations

By using biorthogonal decompositions, we show how uniformly propagating waves, togehter with their velocity, shape, and amplitude, can be extracted from a spatiotemporal signal consisting of the superposition of various traveling waves. The interaction between the different waves manifests itself in space-time resonances in case of a discrete biorthogonal spectrum and in resonant wavepackets in case of a continuous biorthogonal spectrum. Resonances appear as invariant subspaces under the biorthogonal operator, which leads to closed sets of algebraic equations. The analysis is then extended to superpositions of dispersive waves for which the (Fourier) dispersion relation is no longer linear. We then show how a space-time bifurcation, namely a qualitative change in the spatiotemporal nature of the solution, occurs when the biorthogonal operator is a nonholomorphic function of a parameter. This takes place when two eigenvalues are degenerate in the biorthogonal spectrum and when the spatial and temporal eigenvectors rotate within each eigenspace. Such a scenario applied to the superposition of traveling waves leads to the generation of additional waves propagating at new velocities, which can be computed from the spatial and temporal eigenmodes involved in the process (namely the shape of the propagating waves slightly before the bifurcation). An eigenvalue degeneracy, however, does not necessarily lead to a bifurcation, a situation we refer to as being self-avoiding. We illustrate our theoretical predictions by giving examples of bifurcating and self-avoiding events in propagating phenomena.