Showing papers on "Integrable system published in 1994"

Algebro-geometric approach to nonlinear integrable equations

Abstract: A brief but self-contained exposition of the basics of Riemann surfaces and theta functions prepares the reader for the main subject of this text, namely the application of these theories to solving non-linear integrable equations for various physical systems. Physicists and engineers involved in studying solitons, phase transitions or dynamical (gyroscopic) systems, and mathematicians with some background in algebraic geometry and Abelian and automorphic functions, are the targeted audience. This book is suitable for use as a supplementary text to a course in mathematical physics. The authors introduce Riemann surfaces and theta functions as mathematical methods used to analyzse solitons, dynamical systems, phase transitions, etc, and to obtain the solutions of the related non-linear integrable equations.

Quasi-Exactly Solvable Models in Quantum Mechanics

Abstract: QUASI-EXACT SOLVABILITY-WHAT DOES THAT MEAN? Introduction Completely algebraizable spectral problems The quartic oscillator The sextic oscillator Non-perturbative effects in an explicit form and convergent perturbation theory Partial algebraization of the spectral problem The two-dimensional harmonic oscillator Completely integrable quantum systems Deformation of completely integrable models Quasi-exact solvability and the Gaudin model The classical multi-particle Coulomb problem Classical formulation of quantal problems The Infeld-Hull factorization method and quasi-exact solvability The Gelfand-Levitan equation Summary Historical comments SIMPLEST ANALYTIC METHODS FOR CONSTRUCTING QUASI-EXACTLY SOLVABLE MODELS The Lanczos tridiagonalization procedure The sextic oscillator with a centrifugal barrier The electrostatic analogue-the quartic oscillator Higher oscillators with centrifugal barriers The electrostatic analogue-the general case The inverse method of separation of variables The Schrodinger equations with separable variables Multi-dimensional models The "field-theoretical" case Other quasi-exactly solvable models THE INVERSE METHOD OF SEPARATION OF VARIABLES Multi-parameter spectral equations The method-general formulation The case of differential equations Algebraically solvable multi-parameter spectral equations An analytic method Reduction to exactly solvable models The one-dimensional case-classification Elementary exactly solvable models The multi-dimensional case-classification CLASSIFICATION OF QUASI-EXACTLY SOLVABLE MODELS WITH SEPARATE VARIABLE Preliminary comments The one-dimensional non-degenerate case The non-degenerate case-the first type The non-degenerate case-the second type The non-degenerate case-the third type The one-dimensional simplest degenerate case The simplest degenerate case-the first type The simplest degenerate case-the second type The simplest degenerate case-the third type The one-dimensional twice-degenerate case The twice-degenerate-the first type The twice-degenerate case-the second type The one-dimensional most degenerate case The multi-dimensional case COMPLETELY INTEGRABLE GAUDIN MODELS AND QUASI-EXACT SOLVABILITY Hidden symmetries Partial separation of variables Some properties of simple Lie algebras Special decomposition in simple Lie algebras The generalized Gaudin model and its solutions Quasi-exactly solvable equations Reduction to the Schrodinger form Conclusions Appendices A: The Inverse Schrodinger Problem and Its Solution for Several Given States Appendices B: The Generalized Quantum Tops and Exact Solvability Appendices C: The Method of Raising and Lowering Operators Appendices D: Lie Algebraic Hamiltonians and Quasi-Exact Solvability References Index

Group-Theoretical Methods in the Theory of Finite-Dimensional Integrable Systems

Abstract: The present survey is devoted to a general group-theoretic scheme which allows to construct integrable Hamiltonian systems and their solutions in a systematic way. This scheme originates from the works of Kostant [1979a] and Adler [1979] where some special but very instructive examples were studied. Some years later a link was established between this scheme and the so-called classical R-matrix method (Faddeev [1984], Semenov-Tian-Shansky [1983]). One of the advantages of this approach is that it unveils the intimate relationship between the Hamiltonian structure of an integrable system and the specific Riemann problem (or, more generally, factorization problem) that is used to find its solutions. This shows, in particular, that the Hamiltonian structure is completely determined by the Riemann problem. The simplest system which may be studied in this way is the open Toda lattice already described in Chapter 1 by Olshanetsky and Perelomov. (The Toda lattices will be considered here again in a more general framework.) However, the most interesting examples are related to infinite-dimensional Lie algebras. In fact, it can be shown that the solutions of Hamiltonian systems associated with finite-dimensional Lie algebras have a too simple time dependence (roughly speaking, like trigonometric polinomials). By contrast, genuine mechanical problems often lead to more sophisticated (e.g. elliptic or abelian) functions.

The geometry of peaked solitons and billiard solutions of a class of integrable PDE's

Abstract: The purpose of this Letter is to investigate the geometry of new classes of soliton-like solutions for integrable nonlinear equations. One example is the class of peakons introduced by Camassa and Holm [1993] for a shallow water equation. We put this equation in the framework of complex integrable Hamiltonian systems on Riemann surfaces and draw some consequences from this setting. Amongst these consequences, one obtains new solutions such as quasiperiodic solutions,n-solitons, solitons with quasiperiodic background, billiard, andn-peakon solutions and complex angle representations for them. Also, explicit formulas for phase shifts of interacting soliton solutions are obtained using the method of asymptotic reduction of the corresponding angle representations. The method we use for the shallow water equation also leads to a link between one of the members of the Dym hierarchy and geodesic flow onN-dimensional quadrics. Other topics, planned for a forthcoming paper, are outlined.

R‐matrix approach to lattice integrable systems

Abstract: An r‐matrix formalism is applied to the construction of the integrable lattice systems and their bi‐Hamiltonian structure. Miura‐like gauge transformations between the hierarchies are also investigated. In the end the ladder of linear maps between generated hierarchies is established and described.

Drift and diffusion in phase space

Abstract: The problem of stability of action variables (i.e. of the adiabatic invariants) in perturbations of completely integrable (real analytic) hamiltonian systems with more than two degrees of freedom is considered. Extending the analysis of [A], we work out a general quantitative theory, from the point of view of dimensional analysis, for a priori unstable systems (i.e. systems for which the unperturbed integrable part possesses separatrices), proving, in general, the existence of the so-called Arnold's diffusion and establishing upper bounds on the time needed for the perturbed action variables to drift by an amount of O(1). The above theory can be extended so as to cover cases of a priori stable systems (i.e. systems for which separatrices are generated near the resonances by the perturbation). As an example we consider the «d'Alembert precession problem in Celestial Mechanics» (a planet modelled by a rigid rotational ellipsoid with small «flatness» η, resolving on a given Keplerian orbit of eccentricity e=η c , c>1, around a fixed star and subject only to Newtonian gravitational forces) proving in such a case the existence of Arnold's drift and diffusion; this means that there exist initial data for which for any η¬=0 small enough, the planet changes, in due (η-dependent) time, the inclination of the precession cone by an amount of O(1). The homo/heteroclinic angles (introduced in general and discussed in detail together with homoclinic splittings and scatterings) in the d'Alembert problem are not exponentially small with η (in site of first order predictions based upon Melnikov type integrals)

Elliptic Calogero-Moser system from two-dimensional current algebra

Abstract: We show that elliptic Calogero-Moser system and its Lax operator found by Krichever can be obtained by Hamiltonian reduction from the integrable Hamiltonian system on the cotangent bundle to the central extension of the algebra of SL(N,C) currents.Elliptic deformation of Yang-Mills theory is presented.

Surfaces in terms of 2 by 2 matrices. Old and new integrable cases

Abstract: Many of the equations which now are called integrable have been known in differential geometry for a long time. Probably the first was the famous sine-Gordon equation, which was derived to describe surfaces with constant negative Gaussian curvature. At that time many features of integrability of the sine-Gordon and other integrable equations were discovered 1, namely those which have clear geometrical interpretation (for example, the Backlund transform).

New Coupled Integrable Dispersionless Equations

Abstract: New coupled integrable dispersionless equations are presented and solved by the inverse scattering method.

Formation of shape-preserving pulses in a nonlinear adiabatically integrable system

Abstract: We have obtained a class of solitary wave solutions to novel exactly integrable nonlinear wave equations. Conservation laws can be identified and velocities of propagation predicted. We propose to test our predictions in the optical domain with two-color experiments.

A powerful approach to generate new integrable systems

Abstract: In this paper, an effective algorithm to generate integrable systems is given. As a result, many new integrable equations are derived in a systematic way.

Linear r-matrix algebra for classical separable systems

Abstract: We consider a hierarchy of the natural-type Hamiltonian systems of n degrees of freedom with polynomial potentials separable in general ellipsoidal and general paraboloidal coordinates. We give a Lax representation in terms of 2*2 matrices for the whole hierarchy and construct the associated linear r-matrix algebra with the r-matrix dependent on the dynamical variables. A Yang-Baxter equation of dynamical type is proposed. Using the method of variable separation, we provide the integration of the systems in classical mechanics constructing the separation equations and, hence, the explicit form of action variables. The quantization problem is discussed with the help of the separation variables.

Hirota equation as an example of an integrable symplectic map

Abstract: The Hamiltonian formalism is developed for the sine-Gordon model on the spacetime light-like lattice, first introduced by Hirota. The evolution operator is explicitly constructed in the quantum variant of the model and the integrability of the corresponding classical finite-dimensional system is established.

Vector-matrix generalizations of classical integrable equations

Abstract: Some vector-matrix generalizations, both known and new, for well-known integrable equations are presented. All of them possess higher symmetries and conservation laws.

Homoclinic tangles-classification and applications

Abstract: Here we develop the topological approximation method which gives a new description of the mixing and transport processes in chaotic two-dimensional time-periodic Hamiltonian flows. It is based upon the structure of the homoclinic tangle, and supplies a detailed solution to a transport problem for this class of systems, the characteristics of which are typical to chaotic, yet not ergodic dynamical systems. These characteristics suggest some new criteria for quantifying transport and mixing-hence chaos-in such systems. The results depend on several parameters, which are found by perturbation analysis in the near integrable case, and numerically otherwise. The strength of the method is demonstrated on a simple model. We construct a bifurcation diagram describing the changes in the homoclinic tangle as the physical parameters are varied. From this diagram we find special regions in the parameter space in which we approximate the escape rates from the vicinity of the homoclinic tangle, finding nontrivial self-similar solutions as the forcing magnitude tends to zero. We compare the theoretical predictions with brute force calculations of the escape rates, and obtain satisfactory agreement.

Hirota equation as an example of integrable symplectic map

Abstract: The hamiltonian formalism is developed for the sine-Gordon model on the space-time light-like lattice, first introduced by Hirota. The evolution operator is explicitely constructed in the quantum variant of the model, the integrability of the corresponding classical finite-dimensional system is established.

Integrable XYZ spin chain with boundaries

Abstract: We consider a general class of boundary terms of the open XYZ spin-1/2 chain compatible with integrability. We have obtained the general elliptic solution of a K-matrix obeying the boundary Yang-Baxter equation using the R-matrix of the eight-vertex model and derived the associated integrable spin-chain Hamiltonian.

Harmonic maps and integrable systems

Abstract: Introduction and background material the geometry of surfaces sigma and chiral models the algebraic approach the twistor approach.

Finite-gap solutions of the Davey-Stewartson equations

Abstract: We describe the finite-gap solutions of different modifications of the Davey-Stewartson (DS) equations. The restrictions on the spectral data which give us solutions of the real forms DS1 and DS2+ of DS are the same as those in the case of KP1 and KP2 of the Kadomtsev-Petviashvily equation. But for DS2− the restrictions that we regard have no analogues in other integrable systems. We describe also the restrictions that provide regularity of those solutions for DS1 and DS2±. The finite-gap solutions include rational and soliton solutions. We give some classes of those solutions. The well-known dromions for DS1 are examples of that kind.

On the simplest integrable equation in 2+1

Abstract: An integrable two-dimensional generalization of the non-linear Schrodinger equation is discussed. This equation is the simplest scalar evolution equation in two dimensions, which can be integrated by the inverse spectral method. This example is used to illustrate the important fact that to each one-dimensional integrable equation, there correspond several two-dimensional integrable generalizations.

Structure of the conservation laws in integrable spin chains with short range interactions

Abstract: We present a detailed analysis of the structure of the conservation laws in quantum integrable chains of the XYZ-type and in the Hubbard model. With the use of the boost operator, we establish the general form of the XYZ conserved charges in terms of simple polynomials in spin variables and derive recursion relations for the relative coefficients of these polynomials. For two submodels of the XYZ chain - namely the XXX and XY cases, all the charges can be calculated in closed form. For the XXX case, a simple description of conserved charges is found in terms of a Catalan tree. This construction is generalized for the su(M) invariant integrable chain. We also indicate that a quantum recursive (ladder) operator can be traced back to the presence of a hamiltonian mastersymmetry of degree one in the classical continuous version of the model. We show that in the quantum continuous limits of the XYZ model, the ladder property of the boost operator disappears. For the Hubbard model we demonstrate the non-existence of a ladder operator. Nevertheless, the general structure of the conserved charges is indicated, and the expression for the terms linear in the model's free parameter for all charges is derived in closed form.

On the integrability of the inhomogeneous spherically symmetric Heisenberg ferromagnet in arbitrary dimensions

Abstract: The dynamics of an inhomogeneous spherically symmetric continuum Heisenberg ferromagnet in arbitrary (n‐) dimensions is considered. By a known geometrical procedure the spin evolution equation equivalently is rewritten as a generalized nonlinear Schrodinger equation. A Painleve singularity structure analysis of the solutions of the equation shows that the system is integrable in arbitrary (n‐) dimensions only when the inhomogeneity is of inverse power in the radial coordinate in the form f(r)=e1r−2(n−1)+e2r−(n−2). This is confirmed by obtaining the associated Lax pair, Backlund transformation, and the solitonlike solution of the evolution equation. Further, calculations show that the one‐dimensional linearly inhomogeneous ferromagnet acts as a universal model to which all the integrable higher‐dimensional inhomogeneous spherically symmetric spin models can be formally mapped.