On generalized Lotka-Volterra lattices

Generalized matrix Lotka-Volterra lattice equations are obtained in a systematic way from a “master equation” possessing a bicomplex formulation. Presented at the 11th Colloquium “Quantum Groups and Integrable Systems”, Prague, 20–22 June 2002.     

RTT presentation of ortho-symplectic super Yangians. Staggered integrable spin chains

We give a RTT presentation of ortho-symplectic super Yangians that encompasses the orthogonal and symplectic cases. In a second part, we construct an integrable ladder chain model and study its quantum symmetry. Presented at the 11th Colloquium “Quantum Groups and Integrable Systems”, Prague, 20–22 June 2002.     

Quasigraded deformations of loop algebras and classical integrable systems

We construct a family of infinite-dimensional quasigraded Lie algebras, that could be viewed as deformation of the graded loop algebras. Using them we obtain new series of integrable Hamiltonian systems on semisimple Lie algebras and their extensions. Presented at the 11th Colloquium “Quantum Groups and Integrable Systems”, Prague, 20–22 June 2002.     

Twist deformations for algebras of motion

We study the twist deformations of algebras of motiong ^H ⊂ sl(N) with the Cartan subalgebraH(g^H) equal toH(sl(N)). The proposed deformations are maximal in the sense that their carrier algebrasg _c coincide withg ^H. The algebraic properties are demonstrated forg ^H ⊂ sl(5). Presented at the 11th Colloquium “Quantum Groups and Integrable Systems”, Prague, 20–22 June 2002. This work has been partially supported by the Russian Foundation for Fundamental Research under the grant N 00-01-00500.     

Additional Symmetries and String Equation of the CKP Hierarchy

Based on the Orlov and Shulman’s M operator, the additional symmetries and the string equation of the CKP hierarchy are established, and then the higher order constraints on L ^l are obtained. In addition, the generating function and some properties are also given. In particular, the additional symmetry flows form a new infinite dimensional algebra , which is a subalgebra of W _1+∞.      

Operadic deformations as a tool for cogravity

The deformation equation and its integrability condition (Bianchi identity) of a non-(co)associative deformation in operad algebra are found. Based on physical analogies, two ideas ofcogravity equations are proposed. Presented at the 11th Colloquium “Quantum Groups and Integrable Systems”, Prague, 20–22 June 2002.     

On the BKP Hierarchy: Additional Symmetries,Fay Identity and Adler–Shiota–van Moerbeke Formula

We give an alternative proof of the Adler–Shiota–van Moerbeke formula for the BKP hierarchy. The proof is based on a simple expression for the generator of additional symmetries and the Fay identity of the BKP hierarchy.      

Exact solutions of semiclassical non-characteristic Cauchy problems for the sine-Gordon equation

The use of the sine-Gordon equation as a model of magnetic flux propagation in Josephson junctions motivates studying the initial-value problem for this equation in the semiclassical limit in which the dispersion parameter ε tends to zero. Assuming natural initial data having the profile of a moving −2π kink at time zero, we analytically calculate the scattering data of this completely integrable Cauchy problem for all ε>0 sufficiently small, and further we invert the scattering transform to calculate the solution for a sequence of arbitrarily small ε. This sequence of exact solutions is analogous to that of the well-known N-soliton (or higher-order soliton) solutions of the focusing nonlinear Schrödinger equation. We then use plots obtained from a careful numerical implementation of the inverse-scattering algorithm for reflectionless potentials to study the asymptotic behavior of solutions in the semiclassical limit. In the limit ε↓0 one observes the appearance of nonlinear caustics, i.e. curves in space-time that are independent of ε but vary with the initial data and that separate regions in which the solution is expected to have different numbers of nonlinear phases.In the appendices, we give a self-contained account of the Cauchy problem from the perspectives of both inverse scattering and classical analysis (Picard iteration). Specifically, Appendix A contains a complete formulation of the inverse-scattering method for generic L¹-Sobolev initial data, and Appendix B establishes the well-posedness for L^p-Sobolev initial data (which in particular completely justifies the inverse-scattering analysis in Appendix A).     

Anticipative control of switched queueing systems

The relevant dynamics of a queueing process can be anticipated by taking future arrivals into account. If the transport from one queue to another is associated with transportation delays, as it is typical for traffic or productions networks, future arrivals to a queue are known over some time horizon and, thus, can be used for an anticipative control of the corresponding flows. A queue is controlled by switching its outflow between “on” and “off” similar to green and red traffic lights, where switching to “on” requires a non-zero setup time. Due to the presence of both continuous and discrete state variables, the queueing process is described as a hybrid dynamical system. From this formulation, we derive one observable of fundamental importance: the green time required to clear the queue. This quantity allows to detect switching time points for serving platoons without delay, i.e., in a “green wave” manner. Moreover, we quantify the cost of delaying the start of a service period or its termination in terms of additional waiting time. Our findings may serve as a basis for strategic control decisions.     

Lagrangean and Hamiltonian fractional sequential mechanics

The models described by fractional order derivatives of Riemann-Liouville type in sequential form are discussed in Lagrangean and Hamiltonian formalism. The Euler-Lagrange equations are derived using the minimum action principle. Then the methods of generalized mechanics are applied to obtain the Hamilton’s equations. As an example free motion in fractional picture is studied. The respective fractional differential equations are explicitly solved and it is shown that the limitα→1⁺ recovers classical model with linear trajectories and constant velocity. Presented at the 11th Colloquium “Quantum Groups and Integrable Systems”, Prague, 20–22 June 2002.     

Painlevé analysis and exact solutions of the resonant Davey-Stewartson system

It is shown that the resonant Davey-Stewartson (RDS) system can pass the Painlev test. By truncating the Laurent series to a constant level term, a dependent variable transformation is naturally derived, which leads to the bilinear forms of the RDS system. From the bilinear equations, through making suitable assumptions, some new soliton solutions are obtained. Some representative profiles of the solitary waves are graphically displayed including the two-line soliton solution, “Y” soliton solution, “V” soliton solution, solitoff, etc. The solutions might be useful to describe the nonlinear phenomena in Madelung fluids, capillarity fluids, and so on.     

Calculation of critical exponents by self-similar factor approximants

The method of self-similar factor approximants is applied to calculating the critical exponents of the O(N)-symmetric ϕ⁴ theory and of the Ising glass. It is demonstrated that this method, being much simpler than other known techniques of series summation in calculating the critical exponents, at the same time, yields the results that are in very good agreement with those of other rather complicated numerical methods. The principal advantage of the method of self-similar factor approximants is the combination of its extraordinary simplicity and high accuracy.     

A quantum exactly solvable non-linear oscillator with quasi-harmonic behaviour

The quantum version of a non-linear oscillator, previously analyzed at the classical level, is studied. This is a problem of quantization of a system with position-dependent mass of the form m = (1 + λx²)⁻¹ and with a λ-dependent non-polynomial rational potential. This λ-dependent system can be considered as a deformation of the harmonic oscillator in the sense that for λ → 0 all the characteristics of the linear oscillator are recovered. First, the λ-dependent Schrödinger equation is exactly solved as a Sturm-Liouville problem, and the λ-dependent eigenenergies and eigenfunctions are obtained for both λ > 0 and λ < 0. The λ-dependent wave functions appear as related with a family of orthogonal polynomials that can be considered as λ-deformations of the standard Hermite polynomials. In the second part, the λ-dependent Schrödinger equation is solved by using the Schrödinger factorization method, the theory of intertwined Hamiltonians, and the property of shape invariance as an approach. Finally, the new family of orthogonal polynomials is studied. We prove the existence of a λ-dependent Rodrigues formula, a generating function and λ-dependent recursion relations between polynomials of different orders.     

Analysis of the correction factor in the spreading-resistance technique for monitoring epitaxial silicon

The range of applicability of the mixed-boundary-value method for calculating spreading resistance for a homogeneous slab with an effective contact-radius source and backed by a substrate of arbitrary but finite resistivity is investigated. Solutions are presented in terms of the correction factor and the source current density distributions for a slab of varying thickness and with various high-resistivity substrates. The correction factors for different schemes of calculation are correctly obtained by means of introducing an additional current source of opposite sign, but with the same absolute value, located at the drain point contact. This combination of source and drain currents gives the true value for the potential distribution along the surface of the semiconductor. This, in turn, leads to new terms in the equations obtained for the correction factors, which have been omitted in previously published works. A comparison between several schemes of calculation is presented. Within the framework of “uniform” and “variable” current distributions underneath the contact probe, there are two limits for the correction factor. A model based on a combination of these approaches is discussed, and a comparison between the proposed method and the Schumann–Gardner formulation is made. Received: 30 January 2002 / Accepted: 30 January 2002 / Published online: 28 October 2002 RID="*" ID="*"Corresponding author. Fax: +7-095/531-83-54, E-mail: telkom@df.ru     

Three-dimensional motion of a vortex filament and its relation to the localized induction hierarchy

Three-dimensional motion of a slender vortex tube, embedded in an inviscid incompressible fluid, is investigated under the localized induction approximation for the Euler equations. Using the method of matched asymptotic expansions in a small parameter ε, the ratio of core radius to curvature radius, the velocity of a vortex filament is derived to O(ε³), whereby the influence of elliptical deformation of the core due to the self-induced strain is taken into account. It is found that there is an integrable line in the core whose evolution obeys a summation of the first and third terms of the localized induction hierarchy. Received 2 October 2001 / Received in final form 10 May 2002 Published online 2 October 2002 RID="a" ID="a"e-mail: yasuhide@math.kyushu-u.ac.jp     

Finiteness of integrable n-dimensional homogeneous polynomial potentials

We consider natural Hamiltonian systems of n>1$n>1$ degrees of freedom with polynomial homogeneous potentials of degree k. We show that under a genericity assumption, for a fixed k, at most only a finite number of such systems is integrable. We also explain how to find explicit forms of these integrable potentials for small k.     

Comment on “Minimal parabolic quantum groups in twist deformations" by M. Ilyin and V. Lyakhovsky On a “new" deformation of GL(2)

We refute a recent claim in the literature [Czech. J. Phys. 56, 1191 (2006)] of a “new" quantum deformation of GL(2).     

Coarsening process in one-dimensional surface growth models

Surface growth models may give rise to instabilities with mound formation whose typical linear size L increases with time (coarsening process). In one dimensional systems coarsening is generally driven by an attractive interaction between domain walls or kinks. This picture applies to growth models for which the largest surface slope remains constant in time (corresponding to model B of dynamics): coarsening is known to be logarithmic in the absence of noise ( L(t) ∼ ln t) and to follow a power law ( L(t) ∼t ^1/3) when noise is present. If the surface slope increases indefinitely, the deterministic equation looks like a modified Cahn-Hilliard equation: here we study the late stages of coarsening through a linear stability analysis of the stationary periodic configurations and through a direct numerical integration. Analytical and numerical results agree with regard to the conclusion that steepening of mounds makes deterministic coarsening faster : if α is the exponent describing the steepening of the maximal slope M of mounds ( M ^α∼L) we find that L(t) ∼t ⁿ: n is equal to for 1≤α≤2 and it decreases from to for α≥2, according to n = α/(5α - 2). On the other side, the numerical solution of the corresponding stochastic equation clearly shows that in the presence of shot noise steepening of mounds makes coarsening slower than in model B: L(t) ∼t ^1/4, irrespectively of α. Finally, the presence of a symmetry breaking term is shown not to modify the coarsening law of model α = 1, both in the absence and in the presence of noise. Received 28 September 2001 and Received in final form 21 November 2001     

The q-gamma and (q,q)-polygamma functions of Tsallis statistics

An axiomatic definition is given for the q-gamma function of Tsallis (non-extensive) statistical physics, the continuous analogue of the q-factorial of Suyari [H. Suyari, Physica A 368 (1) (2006) 63], and the q-analogue of the gamma function of Euler and Gauss. A working definition in closed form, based on the Hurwitz and Riemann zeta functions (including their analytic continuations), is shown to satisfy this definition. Several relations involving the q-gamma and other functions are obtained. The (q,q)-polygamma functions , defined by successive derivatives of , where ln_qa=(1−q)⁻¹(a^1−q−1),a>0 is the q-logarithmic function, are also reported. The new functions are used to calculate the inferred probabilities and multipliers for Tsallis systems with finite numbers of particles N?∞.     

Pulse propagation over smooth irregular terrain

The source of the electromagnetic field is assumed to be a vertical electric dipole at height ξ above the surface of the plane earth with arbitrary time varying moment. The problem of finding the transient field of this dipole when the earth is allowed to be slightly rough is solved by means of a perturbation analysis, repeated application of integral transforms and their inversion on the base of Cagniard's method with the modification of de Hoop.