Coupled KdV equations with multi-Hamiltonian structures

We present a large class of systems of N-equations which possess (N+1) purely differential, compatible Hamiltonian structures. Our generic equations are isospectral to [(Σ^N−1₀ε_iλ²ⁱ)∂²+Σ^N−1₀υ_iλ²ⁱ]ψ=λ^2Nψ, but our class also includes degenerate, nondispersive systems which are unrelated to this linear problem. Embedded in this class, for each N, there are N distinct coupled KdV systems. When N = 2 our class includes 3 known equations: dispersive water waves, Ito's equation and reduced Benney's equations. These equations are thus tri-Hamiltonian. We also present 2 examples of 3-component quadri-Hamiltonian systems, which generalise the above mentioned dispersive and nondispersive water waves.     

The symmetries of Dynkin diagrams and the reduction of Toda field equations

The exist generalizations of the Toda lattice equations involving the Cartan matrices constructed from the simple and extended root systems of any simple Lie algebra. Toda's original equations correspond to the large-N limit of SU(N). All these equations are known to constitute the integrability conditions for a certain linear problem and as such to have remarkable properties. The symmetries of the equations are investigated by studying the corresponding Dynkin diagrams which conveniently encode the structure of the equations. Corresponding to each conjugacy class of this symmetry group, “reductions” of the equations may be made whereby identification of symmetrically related variables leads to new, self-consistent equations which are integrable in the same sense as before. The new equations which can be regarded as multicomponent generalizations of the Bullough-Dodd equation are shown to correspond precisely to the generalized Cartan matrices classified in the mathematical literature.     

Effects of long-range interaction on critical and multicritical behavior of compressible disordered systems

A field-theoretic approach is applied to describe behavior of weakly disordered, isotropic elastic compressible systems with long-range interactions directly in the three-dimensional space for various values of the long-range interaction parameter a. A renormalization-group procedure is applied separately for a > 2 and a ≤ 2 directly in the three-dimensional space. Renormalization-group equations are analyzed in the two-loop approximation, and critical and tricritical points are determined. It is shown that long-range effects are not important when a ≤ 2, whereas they play a key role in the opposite case of a > 2. Critical exponents characterizing the system are obtained for various values of the long-range interaction parameter. Behavior of homogeneous and disordered systems characterized by two fluctuating order parameters is also described.     

Quark-antiquark composite systems: The Bethe-Salpeter equation in the spectral-integration technique in the case of different quark masses

The Bethe-Salpeter equations for quark-antiquark composite systems with different quark masses, such as \(q\bar s(with q = u,d),q\bar Q\), and \(s\bar Q\) (with Q = c, b), are written in terms of spectral integrals. For mesons characterized by the mass M, spin J, and radial quantum number n, the equations are written for the (n, M²) trajectories with fixed J. The mixing between states with different quark spin S and angular momentum L is also discussed.     

Lie-algebra contractions and separation of variables. Three-dimensional sphere

The Inönü-Wigner contraction from the SO(4) group to the Euclidean E(3) group is used to relate the separation of variables in Helmholtz equations for two corresponding homogeneous spaces. We show how the six systems of coordinates on the three-dimensional sphere contracted to nine systems of coordinates on Euclidean space. As a consequence of the Inönü-Wigner contraction we also consider contractions of the integrals of motion.     

Lie algebra contractions on two-dimensional hyperboloid

The Inönü-Wigner contraction from the SO(2, 1) group to the Euclidean E(2) and E(1, 1) group is used to relate the separation of variables in Laplace-Beltrami (Helmholtz) equations for the four corresponding two-dimensional homogeneous spaces: two-dimensional hyperboloids and two-dimensional Euclidean and pseudo-Euclidean spaces. We show how the nine systems of coordinates on the two-dimensional hyperboloids contracted to the four systems of coordinates on E ₂ and eight on E _1,1. The text was submitted by the authors in English.     

Thermodynamics of metal-insulator systems: the two-fluid model in the presence of a magnetic field

The phenomenological two-fluid model involving localized and itinerant electrons has been successful in explaining zero-field thermodynamic measurements in disordered metal-insulator systems such as Si:P. We have extended this model to include the effects of a magnetic fieldB, and have used the modified equations to account for recent specific heat and susceptibility measurements made as a function ofB. Predictions of the Wilson ratio behaviour at low temperatures in finite fields remain to be tested.     

Symmetry Breaking in Laughlin’s State on a Cylinder

We investigate Laughlin’s fractional quantum Hall effect wave function on a cylinder. We show that it displays translational symmetry breaking in the axial direction for sufficiently thin cylinders. At filling factor 1/p, the period is p times the period of the filled lowest Landau level. The proof uses a connection with one-dimensional polymer systems and discrete renewal equations.     

The equations of motion of macroscopic bodies in general relativity

A new approach to the problem of motion in General Relativity, based upon the systematic approximation procedure of Synge, is presented. The equations of transnational motion for a system of spherical bodies moving under their mutual gravitational attractions are derived. Approximations are based upon the weakness of the field and on the distance between any two of the bodies being considered large by comparision with their radii. The most general stress distribution consistent with maintaining the symmetry of the bodies throughout the motion is chosen. The use of controlled errors enables us to derive equations of motion applicable to a wider class of physical systems than the original equations of Einstein, Infeld and Hoffmann and Fock-Papapetrou.     

Far-from-equilibrium quantum many-body dynamics

A theory of real-time quantum many-body dynamics is evaluated in detail. It is based on a generating functional of correlation functions where the closed time contour extends only to a given time. Expanding the contour from this time to a later time leads to a dynamic flow of the generating functional. This flow describes the dynamics of the system and has an explicit causal structure. In the present work it is evaluated within a vertex expansion of the effective action leading to time-evolution equations for Green functions. These equations are applicable for strongly interacting systems as well as for studying the late-time behavior of non-equilibrium time evolution. For the specific case of a bosonic $\mathcal{N}$ -component φ ⁴-theory with contact interactions an s-channel truncation is identified to yield equations identical to those derived from the 2PI effective action in next-to-leading order of a $1/\mathcal{N}$ expansion. The presented approach allows to directly obtain non-perturbative dynamic equations beyond the widely used 2PI approximations.     

Few Body Systems Made of Pseudoscalars

Few body systems made of pseudoscalars, like ${K\, K \, \bar K,\,\pi \, K \, \bar K}$ , are studied within a coupled channel approach based on solving the Faddeev equations considering two-body chiral t-matrices as input. As a result, we have found dynamical generation of several states which can be associated with some of the pseudoscalar states listed by the Particle Data Group, like K(1460) or π(1300). The amplitudes obtained have been then used to study systems like f ₀(980) π π and ${f_0(980)K \, \bar K}$ and an evidence for a f ₀ resonance around 1,790 MeV is found.     

Collision of Rydberg atom A** with atom B in the ground electronic state. Optical potential

A method for calculating the complex optical potential of slowly colliding Rydberg atom A^** and neutral atom B in the ground electronic state is suggested. The method is based on the asymptotic approach and the theory of multichannel quantum defects, which uses the formalism of renormalized Lippmann-Schwinger equations. The potential is introduced as the 〈q|V _opt|q〉 matrix element of the optical interaction operator, for which the integral equation is derived, and is calculated in the basis set of free particle wave functions |q〉. Fairly simple equations for the shift and broadening of the ionic term are obtained, and the principal characteristics of these equations are analyzed. By way of illustration, the optical potential of the Na^**(nl)+B systems, where B is a rare gas atom, is calculated.     

SOME GENERAL EFFECTS OF STRONG HIGH-FREQUENCY EXCITATION: STIFFENING, BIASING AND SMOOTHENING

Mechanical high-frequency (HF) excitation provides a working principle behind many industrial and natural applications and phenomena. This paper concerns three particular effects of HF excitation, that may change the apparent characteristics of mechanical systems: (1) stiffening, by which the apparent linear stiffness associated with an equilibrium is changed, along with derived quantities such as stability and natural frequencies; (2) biasing by which the system is biased towards a particular state, static or dynamic, which does not exist or is unstable in the absence of the HF excitation; and (3) smoothening, referring to a tendency for discontinuities to be effectively “smeared out” by HF excitation. Illustrating first these effects for a few specific systems, analytical results are provided that quantify them for a quite general class of mechanical systems. This class covers systems that can be modelled by a finite number of second order ordinary differential equations, generally non-linear, with periodically oscillating excitation terms of high frequency and small amplitude. The results should be useful for understanding the effects in question in a broader perspective than is possible with specific systems, for calculating effects for specific systems using well-defined formulas, and for possibly designing systems that display prescribed characteristics in the presence of HF excitation.     

Percolation Thresholds and Phase Transitions in Binary Composites

The metal–insulator and metal–superconductor phase transitions related to the percolation thresholds in two-component composites are considered. The behavior of effective conductivity σ _e in the vicinity of both thresholds is described in terms of the similarity hypothesis. A one-to-one correspondence between the equations derived for σ _e in both critical regions is found for randomly heterogeneous systems.     

Integrable deformations of Lotka-Volterra systems

The Hamiltonian structure of a class of three-dimensional (3D) Lotka-Volterra (LV) equations is revisited from a novel point of view by showing that the quadratic Poisson structure underlying its integrability structure is just a real three-dimensional Poisson-Lie group. As a consequence, the Poisson coalgebra map Δ(2) that is given by the group multiplication provides the keystone for the explicit construction of a new family of 3N-dimensional integrable systems that, under certain constraints, contain N sets of deformed versions of the 3D LV equations. Moreover, by considering the most generic Poisson-Lie structure on this group, a new two-parametric integrable perturbation of the 3D LV system through polynomial and rational perturbation terms is explicitly found.     

Two-dimensional exactly and completely integrable dynamical systems

An investigation of two-dimensional exactly and completely integrable dynamical systems associated with the local part of an arbitrary Lie algebra \(\mathfrak{g}\) whose grading is consistent with an arbitrary integral embedding of 3d-subalgebra in \(\mathfrak{g}\) has been carried out. We have constructed in an explicit form the corresponding systems of nonlinear partial differential equations of the second order and obtained their general solutions in the sense of a Goursat problem. A method for the construction of a wide class of infinite-dimensional Lie algebras of finite growth has been proposed.     

A study of Boltzmann energy equations

The formulation, theory, and numerical resolution of physical Boltzmann energy equations are considered. These equations model the time evolution of the energy distribution function for spatially homogeneous isotropic systems of identical particles. Precise statements of the conservation laws and the H-theorem are made. The possibility of solving such equations in Hilbert space settings is considered both from a theoretical and from a computational viewpoint. The structures of the kernels for a variety of physical models are analyzed in detail, and point. The structures of the kernels for a variety of physical models are analyzed in detail, and a new class of models is proposed which is simple yet has all the right physical features. Numerical results show that enhancement phenomena probably occur for a range of different models. In an appendix the connection through an Abel transform for the Bobylev, Krook, and Wu (BKW) and Tjon and Wu (TW) equations is demonstrated by direct integration.     

Dirac constraint analysis and symplectic structure of anti-self-dual Yang-Mills equations

We present the explicit form of the symplectic structure of anti-self-dual Yang-Mills (ASDYM) equations in Yang’s J- and K-gauges in order to establish the bi-Hamiltonian structure of this completely integrable system. Dirac’s theory of constraints is applied to the degenerate Lagrangians that yield the ASDYM equations. The constraints are second class as in the case of all completely integrable systems which stands in sharp contrast to the situation in full Yang-Mills theory. We construct the Dirac brackets and the symplectic 2-forms for both J- and K-gauges. The covariant symplectic structure of ASDYM equations is obtained using the Witten-Zuckerman formalism. We show that the appropriate component of the Witten-Zuckerman closed and conserved 2-form vector density reduces to the symplectic 2-form obtained from Dirac’s theory. Finally, we present the Bäcklund transformation between the J- and K-gauges in order to apply Magri’s theorem to the respective two Hamiltonian structures.     

Noise-Induced Drift in Stochastic Differential Equations with Arbitrary Friction and Diffusion in the Smoluchowski-Kramers Limit

We consider the dynamics of systems with arbitrary friction and diffusion. These include, as a special case, systems for which friction and diffusion are connected by Einstein fluctuation-dissipation relation, e.g. Brownian motion. We study the limit where friction effects dominate the inertia, i.e. where the mass goes to zero (Smoluchowski-Kramers limit). Using the Itô stochastic integral convention, we show that the limiting effective Langevin equations has different drift fields depending on the relation between friction and diffusion. Alternatively, our results can be cast as different interpretations of stochastic integration in the limiting equation, which can be parametrized by α∈?. Interestingly, in addition to the classical Itô (α=0), Stratonovich (α=0.5) and anti-Itô (α=1) integrals, we show that position-dependent α=α(x), and even stochastic integrals with α?[0,1] arise. Our findings are supported by numerical simulations.