Slaving principle for stochastic differential equations with additive and multiplicative noise and for discrete noisy maps

We first treat multidimensional nonlinear noisy maps. We assume that the variables can be split into two classes of variablesu ands so that the linearized equations would give rise to growth or decay foru ands, respectively. We show how the slaved variabless can be explicitly expressed by the order parametersu by making use of the fully nonlinear equations. By taking the limit of vanishing time steps and using a Wiener process and the Îto calculus we derive the corresponding formulas for stochastic differential equations (including multiplicative noise). In this way a high-dimensional problem can be reduced to a problem of much lower dimensions described again by stochastic equations of theÎto type. A similar procedure holds for theStratonovich calculus.     

Field Theory Describing Arbitrary Two-Body Scattering Amplitude as a Born Term

We propose in this paper the local relativistic quantum field-theoretic formulation of the most general isobar model, with unstable two-particle isobars having arbitrary mass and spin spectrum. Such infinite family of isobars is described by infinite-component field operator with its components given by symmetric, traceless and divergenceless field operators ϕu₁…uI (x;s), where I describes spin (I = 0, 1, 2…) and the continuous parameter S the spectrum of asymptotic masses. The asymptotic free states defined by generalized LSZ limits of ϕu₁…uI (x;s), can be described by means of stable multiparticle states. In such a field theory one can choose the parameters in such a way that already a Born term describes an arbitrarily chosen two-body scattering amplitude. As an example we present the generalization of Van Hove model of Reggeons to the case of complex Regge trajectory, describing Regge family of resonances on the second sheet. Finally the correspondence with conventional theory is discussed. It is shown that the Feynman diagrams in local field-theoretic isobar model can be interpreted in the framework of non local theory describing in some approximation an equivalent interaction between stable decay products.     

Ergodic Theory of Parabolic Horseshoes

In this paper we develop the ergodic theory for a horseshoe map f which is uniformly hyperbolic, except at one parabolic fixed point ω and possibly also on W ^s (ω). We call f a parabolic horseshoe map. In order to analyze dynamical and geometric properties of such horseshoes, by making use of induced maps, we establish, in the context of σ-finite measures, an appropriate version of the variational principle for continuous potentials with mild distortion defined on subshifts of finite type. Staying in this setting, we propose a concept of σ-finite equilibrium states (each classical probability equilibrium state is a σ-finite equilibrium state). We then study the unstable pressure function , the corresponding finite and σ-finite equilibrium states and their associated conditional measures. The main idea is to relate the pressure function to the pressure of an embedded parabolic iterated function system and to apply the developed theory of the symbolic σ-finite thermodynamic formalism. We prove, in particular, an appropriate form of the Bowen-Ruelle-Manning-McCluskey formula, the existence of exactly two σ-finite ergodic conservative equilibrium states for the potential –t ^u log |Df|E ^u | (where t ^u denotes the unstable dimension), one of which is the Dirac δ-measure supported at the parabolic fixed point and the other being non-atomic. We also show that the conditional measures of this non-atomic equilibrium state on unstable manifolds, are equivalent to (finite and positive) packing measures, whereas the Hausdorff measures vanish. As an application of our results we obtain a classification for the existence of a generalized physical measure, as well as a criteria implying the non-existence of an ergodic measure of maximal dimension. The research of the first author was supported in part by the NSF Grant DMS 0400481. The research of the second author was supported in part by the National Science Foundation under Grant No. EPS-0236913 and matching support from the State of Kansas through Kansas Technology Enterprise Corporation.     

A generalized Weyl—Wigner quantization scheme unifying P-Q and Q-P ordering and Weyl ordering of operators

By extending the usual Wigner operator to the s-parameterized one as 1/4π2 integral (dyduexp [iu(q-Q)+iy(p-P)+is/2yu]) from n=- ∞ to ∞ with s beng a,real parameter,we propose a generalized Weyl quantization scheme which accompanies a new generalized s-parameterized ordering rule.This rule recovers P-Q ordering,Q-P ordering,and Weyl ordering of operators in s = 1,1,0 respectively.Hence it differs from the Cahill-Glaubers’ ordering rule which unifies normal ordering,antinormal ordering,and Weyl ordering.We also show that in this scheme the s-parameter plays the role of correlation between two quadratures Q and P.The formula that can rearrange a given operator into its new s-parameterized ordering is presented.     

Surface instability of a current carrying plasma under the influence of a high frequency electric field

Surface instability of a collisionless semi-infinite current carrying plasma is studied. The semi-infinite plasma bounded by a plane surface is under the influence of a high frequency (hf) field. There are two classes of surface modes. One is a normal extension of zero high frequency field and the other due entirely to the presence ofhf field. As expected, with the increase in thehf field, the growth rates of the surface instabilities decrease. There are regions defined by the electron drift velocityu where the unstable surface and bulk regions overlap. The interesting result is that unlike the bulk plasma, there is a stable region on theu-axis flanked by two unstable regions. The width of this stable region increases with the increase in the field strength.     

Application of the slaving principle to quenching problems

The slaving principle theory is applied to quenching problems. It is shown that a system, which starts from the close vicinity of an unstable point and the evolution of which is described by a set ofn-dimensional equations, will approach and then follow the unstable manifold. In the presence of fluctuations, it is even possible that the system approaches the most unstable manifold. Thus, the actual dimension related to the problem can be considerably reduced.     

The Rate of Convergence of Fourier Coefficients for Entire Functions of Infinite Order with Application to the Weideman-Cloot Sinh-Mapping for Pseudospectral Computations on an Infinite Interval

We analytically compute the asymptotic Fourier coefficients for several classes of functions to answer two questions. The numerical question is to explain the success of the Weideman-Cloot algorithm for solving differential equations on an infinite interval. Their method combines Fourier expansion with a change-of-coordinate using the hyperbolic sine function. The sinh-mapping transforms a simple function like exp(-z²) into an entire function of infinite order. This raises the second, analytical question: What is the Fourier rate of convergence for entire functions of an infinite order? The answer is: Sometimes even slower than a geometric series. In this case, the Fourier series converge only on the real axis even when the function u (z) being expanded is free of singularities except at infinity. Earlier analysis ignored stationary point contributions to the asymptotic Fourier coefficients when u(z) had singularities off the real z-axis, but we show that sometimes these stationary point terms are more important than residues at the poles of u(z).     

The lie algebra of invariant group of the KdV,MKdV, or Burgers equation

Let (E): u _t=H(u) denote the KdV, MKdV or Burgers equation, and U(s)=(D^j _s)/u _j, where D=d/dx, u _i=Dⁱ _u, s=s(u, u ₁, ..., u _n) is a polynomial of u _i with constant coefficients, be the generator of invariant group of equation (E). We prove in this paper that all such generators form a commutative Lie algebra, from which it follows that for any symmetry s(u, ..., u _n) of (E), the evolution equation u _t=s(u, ..., u _n) possesses an infinite number of symmetries (or conservation laws in the case of KdV and MKdV equations).     

Dislocational kinetics in the intense deformation of fcc single crystals

The stability of the dislocational subsystem of fcc single crystals deformed in dynamic conditions is investigated. It is shown that, depending on the deformational conditions, the system may have one or two steady states, one of which is ρ _s ⁽¹⁾ . When the system has one trivial steady state, it may be stable or unstable. In some conditions, a second unstable point ρ _s ⁽²⁾ appears; in this case, ρ _s ⁽¹⁾ is stable. Tomsk State Architectural and Building Academy. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 8, pp. 43–48, August, 1997.     

Superconducting instability in the large-U limit of the two-dimensional Hubbard model

We have investigated numerically the pairing instabilities of Gutzwiller wavefunctions. These are equivalent to a certain form of the resonant valence bond wavefunction. The case considered is a nearly half-filled two dimensional band with interactions given by a Hubbard model with large on-site Coulomb interactions. We find that the paramagnetic normal state is unstable tod-wave pairing but stable againsts-wave pairing. The antiferromagnetic state is marginally stable against both types of pairing. These results can be explained as an interference effect resulting in enhanced antiferromagnetic spin correlation in the paired state.     

On the well-posedness of the Holm-Staley b-family of equations

In this paper we consider the Holm-Staley b-family of equations in the Sobolev spaces H^s(?) for s>3/2. Using a geometric approach we show that, for any value of the parameter b, the corresponding solution map,u(0)?u(T ), is nowhere locally uniformly continuous.     

Small BGK Waves and Nonlinear Landau Damping

Consider a 1D Vlasov-poisson system with a fixed ion background and periodic condition on the space variable. First, we show that for general homogeneous equilibria, within any small neighborhood in the Sobolev space ${W^{s,p}\left( p >1 ,s <1 +\frac{1}{p}\right)}${W^{s,p}\left( p >1 ,s <1 +\frac{1}{p}\right)} of the steady distribution function, there exist nontrivial travelling wave solutions (BGK waves) with arbitrary minimal period and traveling speed. This implies that nonlinear Landau damping is not true in W^s,p( s < 1 +\frac1p){W^{s,p}\left( s <1 +\frac{1}{p}\right)} space for any homogeneous equilibria and any spatial period. Indeed, in a W^s,p(s < 1 +\frac1p){W^{s,p}\left(s <1 +\frac{1}{p}\right)} neighborhood of any homogeneous state, the long time dynamics is very rich, including travelling BGK waves, unstable homogeneous states and their possible invariant manifolds. Second, it is shown that for homogeneous equilibria satisfying Penrose’s linear stability condition, there exist no nontrivial travelling BGK waves and unstable homogeneous states in some ${W^{s,p}\left( p >1 ,s >1 +\frac{1}{p}\right)}${W^{s,p}\left( p >1 ,s >1 +\frac{1}{p}\right)} neighborhood. Furthermore, when p = 2, we prove that there exist no nontrivial invariant structures in the ${H^{s}\left( s > \frac{3}{2}\right) }${H^{s}\left( s > \frac{3}{2}\right) } neighborhood of stable homogeneous states. These results suggest the long time dynamics in the ${W^{s,p}\left( s >1 +\frac{1}{p}\right) }${W^{s,p}\left( s >1 +\frac{1}{p}\right) } and particularly, in the ${H^{s}\left( s > \frac{3}{2}\right) }${H^{s}\left( s > \frac{3}{2}\right) } neighborhoods of a stable homogeneous state might be relatively simple. We also demonstrate that linear damping holds for initial perturbations in very rough spaces, for a linearly stable homogeneous state. This suggests that the contrasting dynamics in W ^{s, p} spaces with the critical power s=1+\frac1p{s=1+\frac{1}{p}} is a truly nonlinear phenomena which can not be traced back to the linear level.     

Infrared singularities in one-dimensional fermi systems

We study the infrared problem in one-dimensional Fermi systems with general interactions including backward scattering. The singular structure of the ground state energy which is a function of two coupling parametersu andv is determined exactly. For attractiveu-coupling one has power law singularities inv with a characteristic singular exponentk varying continuously withu. This behavior is reminiscent of a phase transition of continuous order and is obtained here by exploiting the analogy of the 1-d Fermi system and a 2-d classical plasma with a long range logarithmic Coulomb potential and a short range cut off. We also discuss other recent approaches to the problem.Supported in part by the Army Research Office, Durham, N.C., while the author was a summer visitor at Stanford University.     

Laser-induced fluorescence of Na2 at the Na(3s → 5s) two-photon transition (6022.3 Å)

Na₂ excited from the X¹Σ_g⁺ state to the A¹Σ_u⁺ state by a narrow band (3 MHz) Rhodamine-6G dye laser at 6022.3 Å, the same wavelength at which Na undergoes the 3s–5s two-photon transition, gives four fluorescence series from A¹Σ_u⁺ levels (v′ = 21, J′ = 26), (18, 33), (33, 19), and (34, 50). The last two series are much weaker in intensity, and at long wavelengths many doublets are lost in the background noise. The same (34, 50) fluorescence series was found by other workers in the lab using a Kr⁺ (5682 Å) laser as excitation source. Their analysis agrees very well with the findings in the work.     

Partial Wave Crossing Relations for Meson-Baryon Scattering

Three different types of relations for meson-baryon partial wave amplitudes following from the s ↔ u crossing symmetry of the invariant amplitudes are investigated: i) the partial wave crossing relations directly derived from the s ↔ u crossing symmetry and the partial wave projection, ii) the partial wave crossing relations following from a combined use of s ↔ u crossing symmetry and analyticity, and iii) the crossing sum rules, which are a direct consequence of the s ↔ u crossing symmetry. The kernels entering the crossing relations i) and ii) are given in an analytic form, valid for arbitrary angular momentum. Their properties are studied in detail. By means of the partial wave crossing relations ii) it is demonstrated that u-channel exchange of a resonance leads to a resonance-like behaviour of the partial wave amplitudes on the crossed physical cut. This phenomenon is described in terms of “pseudoresonances”. Finally the crossing sum rules iii) and their practical use are thoroughly discussed, and they are compared with the partial wave crossing relations i) and ii). Special attention is paid to the crossing symmetry constraints on the S-wave amplitudes. As a result of these investigations we present a crossing symmetric ansatz for these amplitudes which gives a more reliable parametrization than the usual effective range approximation.     

More on Zeta-Function Regularization of High-Temperature Expansions

A recent paper using the Riemann ζ-function to regularize the (divergent) coefficients occurring in the high-temperature expansions of one-loop thermodynamic potentials is extended. This method proves to be a powerful tool for converting Dirichlet-type series Σ_ma_m(x_i)/m^s into powerseries in the dimensionless parameters x_i. The coefficients occurring in the power series are (proportional to) ζ-functions evaluated away from their poles - this is where the regularization occurs. High-temperature expansions are just one example of this highly-nontrivial rearrangement of Dirichlet series into power series form. We discuss in considerable detail series in which a_m(x_i) is a product of trigonometrie, algebraic and Bessel function factors. The ζ-function method is carefully explained, and a large number of new formulae are provided. The means to generalize these formulae are also provided. Previous results on thermodynamic potentials are generalized to include a nonzero constant term in the gauge potential (time component) which can be used to probe the electric sector of temperature gauge theories.     

An estimation of pollutant diffusion rates during calm conditions

Summary Diffusion rates during calm conditions have been determined in dependence of stability. Similarity laws in an unstable and neutral atmosphere can be applied to calms provided the wind velocityu is substituted by the ?critical? wind velocityu _c which defines the onest of a calm. For a stable stratification vertical diffusion rates may be estimated by means ofP-G curves and the same procedure was applied to the horizontal diffusion as well. Paper presented at the GNFAO/EURASAP Meeting, Turin, September 1989. To speed up publication, proofs were not sent to the authors and were supervised by the Scientific Committee.     

Aeolian transport of sand

The airborne transport of particles on a granular surface by the saltation mechanism is studied through numerical simulation of particles dragged by turbulent air flow. We calculate the saturated flux q_s and show that its dependence on the wind strength u_* is consistent with several empirical relations obtained from experimental measurements. We propose and explain a new relation for fluxes close to the threshold velocity u_t, namely, q_s=a(u_*-u_t)^α with α≈2. We also obtain the distortion of the velocity profile of the wind due to the drag of the particles and find a novel dynamical scaling relation. We also obtain a new expression for the dependence of the height of the saltation layer as function of the strength of the wind.     

Eigenvalues and eigenfunctions of a clover plate

We report a numerical study of the flexural modes of a plate using semi-classical analysis developed in the context of quantum systems. We first introduce the Clover billiard as a paradigm for a system inside which rays exhibit stable and chaotic trajectories. The resulting phase space explored by the ray trajectories is illustrated using the Poincare surface of section, and shows that it has both integrable and chaotic regions. Examples of the stable and the unstable periodic orbits in the geometry are presented. We numerically solve the biharmonic equation for the flexural vibrations of the Clover shaped plate with clamped boundary conditions. The first few hundred eigenvalues and the eigenfunctions are obtained using a boundary elements method. The Fourier transform of the eigenvalues show strong peaks which correspond to ray periodic orbits. However, the peaks corresponding to the shortest stable periodic orbits are not stronger than the peaks associated with unstable periodic orbits. We also perform statistics on the obtained eigenvalues and the eigenfunctions. The eigenvalue spacing distribution P(s) shows a strong peak and therefore deviates from both the Poisson and the Wigner distribution of random matrix theory at small spacings because of the C_4v symmetry of the Clover geometry. The density distribution of the eigenfunctions is observed to agree with the Porter-Thomas distribution of random matrix theory. Received 12 February 2001 and Received in final form 17 April 2001     

Nonlinear stability of solitary waves of a generalized Kadomtsev-Petviashvili equation

We prove that the set of solitary wave solutions of a generalized Kadomtsev-Petviashvili equation in two dimensions, (u _t+(u^m+1)_x+u_xxx)_x=u_yy is stable for 0<m<4/3.