arXiv subjects
Publications and source records attributed to K. L. Vaninsky.
In this short survey, we describe our approach for constructing hierarchies of Poisson brackets for classical integrable systems using its' spectral curves.
We establish a new representation of the infinite hierarchy of Pois- son brackets (PB) for the open Toda lattice in terms of its spectral curve. For the classical Poisson bracket (PB) we give a representation in the form of a contour integral of some special Abelian differential (meromorphic one-form) on the spectral curve. All higher brackets of the infinite hierarchy are obtained by multiplication of the one-form by a power of the spectral parameter.
We consider a hierarchy of Poisson structures defined on rational functions on the Riemann sphere. This hierarchy is originated in the theory of the integrable Camassa-Holm equation associated with the Krein's string spectral problem. Previously the proof of Jacobi identity was obtained by reducing the bracket to canonical Darboux coordinates. The main result of this note is a direct proof of the Jacobi identity. It turns out that the direct proof of the Jacobi identity is far from trivial. We also give an example of another hierarchy of Poisson brackets and construct Darboux coordinates for it.
This paper consists of two parts. The first part is devoted to empirical analysis of consolidated order book (COB) for the index RTS futures. In the second part we consider Poissonian multi--agent model of the COB. By varying parameters of different groups of agents submitting orders to the book we are able to model various real life phenomenons. In particular we model the spread, the profile of the book and large price changes. Two different mechanisms of large price changes are considered in detail. One is the disbalance of liquidity in the COB and another is the disbalance of sell and buy orders in the order flow.
Contrary to the theory of Markov processes, no general theory exists for the so called nonlinear Markov processes. We study an example of "nonlinear Markov process" related to classical probability theory, merely to random walks. This model provides interesting phenomena (absent in classical Markov chains): continuum of stationary measures, conserved quantities, convergence to stationary classical random walks etc.
This is the last in a series of four papers on Poisson formalism for the cubic nonlinear Schrodinger equation with repulsive nonlinearity. In this paper we consider scattering potentials.
We consider the integrable Camassa--Holm hierarchy on the line with positive initial data rapidly decaying at infinity. It is known that flows of the hierarchy can be formulated in a Hamiltonian form using two compatible Poisson brackets. In this note we propose a new approach to Hamiltonian theory of the CH equation. In terms of associated Riemann surface and the Weyl function we write an analytic formula which produces a family of compatible Poisson brackets. The formula includes an entire function $f(z)$ as a parameter. The simplest choice $f(z)=1$ or $f(z)=z$ corresponds to the rational or trigonometric solutions of the Yang-Baxter equation and produces two original Poisson brackets. All other Poisson brackets corresponding to other choices of the function $f(z)$ are new.
This is the second in a series of papers on Poisson formalism for the cubic nonlinear Schrödinger equation with repulsive nonlinearity. In this paper we consider periodic potentials. The inverse spectral problem for the periodic auxiliary Dirac operator leads to a hyperelliptic Riemann surface $\G$. Using the spectral problem we introduce on this Riemann surface a meromorphic function $¶$. We call it the Weyl function, since it is closely related to the classical Weyl function discussed in the first paper. We show that the pair $(\G,¶)$ carries a natural Poisson structure. We call it the deformed Atiyah--Hitchin bracket. The Poisson bracket on the phase space is the image of the deformed Atiyah--Hitchin bracket under the inverse spectral transform.
We consider the integrable Camassa--Holm equation on the line with positive initial data rapidly decaying at infinity. On such phase space we construct a one parameter family of integrable hierarchies which preserves the mixed spectrum of the associated string spectral problem. This family includes the CH hierarchy. We demonstrate that the constructed flows can be interpreted as Hamiltonian flows on the space of Weyl functions of the associated string spectral problem. The corresponding Poisson bracket is the Atiyah--Hitchin bracket. Using an infinite dimensional version of the Jacobi ellipsoidal coordinates we obtain a one parameter family of canonical coordinates linearizing the flows.
We develop a unified approach for construction of symplectic forms for 1D integrable equations with the periodic and rapidly decaying initial data. As an example we consider the cubic nonlinear Schrödinger equation.
We develop algebro-geometrical approach for the open Toda lattice. For a finite Jacobi matrix we introduce a singular reducible Riemann surface and associated Baker-Akhiezer functions. We provide new explicit solution of inverse spectral problem for a finite Jacoby matrix. For the Toda lattice equations we obtain the explicit form of the equations of motion, the symplectic structure and Darboux coordinates. We develop similar approach for 2D open Toda. Explaining some the machinery we also make contact with the periodic case.
An invariant Gibbs' state for the nonlinear Schrodinger equation on the circle was constructed by Bourgain, and McKean, out of the basic Hamiltonian using a trigonometric cut-off. The cubic nonlinear Schrodinger equation is a completely integrable system having an infinite number of additional integrals of motion. In this paper we construct the second invariant Gibbs' state from one of these additional integrals for the cubic NLS on the circle. This additional Gibbs' state is singular with respect to the Gibbs' state previously constructed from the basic Hamiltonian. Our approach employs the Ablowitz-Ladik system, a completely integrable discretization of the cubic Schrodinger equation.
The system of $N$ classical particles on the line with the Weierstrass $\wp$ function as potential is known to be completely integrable. Recently D'Hoker and Phong found a beautiful parameterization by the polynomial of degree $N$ of the space of Riemann surfaces associated with this system. In the trigonometric limit of the elliptic potential these Riemann surfaces degenerate into rational curves. The D'Hoker-Phong polynomial in the limit describes the intersection points of the rational curves. We found an explicit determinant representation of the polynomial in the trigonometric case. We consider applications of this result to the theory of Toeplitz determinants and to geometry of the spectral curves. We also prove our earlier conjecture on the asymptotic behavior of the ratio of two symplectic volumes when the number of particles tends to infinity.
We consider classical particles on the line with the Weierstrass $\wp$ function as potential. This system parameterizes special solutions of the KP equation. We derive the trace formula which relates the Hamiltonian of the particle system to the residues of some Abelian differential (meromorphic one-form) on the spectral curve. Such formula is important for the construction action-angle variables and study invariant Gibbs' states.
We consider various trace formulas for the cubic Schrodinger equation in the space of infinitely smooth functions subject to periodic boundary conditions. The formulas relate conventional integrals of motion to the periods of some Abelian differentials (holomorphic one-forms) on the spectral curve. We show that the periods of Abelian differentials are global coordinates on the moduli space of spectral curves. The exterior derivatives of the holomorphic one-forms are the basic and higher symplectic structures on the phase space. We write explicitly these symplectic structures in $QP$ coordinates. We compute the ratio of two symplectic volume elements in the infinite genus limit.
We present two independent approaches for computing the thermodynamics for classical particles interacting via the Moser--Calogero potential. Combining the results we propose the form of equation of state or, what is equivalent, the asymptotics of the Jacobian between volume elements corresponding two symplectic structures on the phase space.
The Dirac operator enters into zero curvature representation for the cubic nonlinear Schrödinger equation. We introduce and study a conformal map from the upper half-plane of the spectral parameter of the Dirac operator into itself. The action variables turn out to be limiting boundary values of the imaginary part of this map. We describe the image of the momentum map (convexity theorem) in the simplest case of a potential from the Schwartz class. We apply this description to the invariant manifolds for the nonlinear Schrödinger equation.