Quantum hydrodynamics from large-<Emphasis Type="Italic">n</Emphasis> supersymmetric gauge theories

We study the connection between periodic finite-difference Intermediate Long Wave (\(\Delta \hbox {ILW}\)) hydrodynamical systems and integrable many-body models of Calogero and Ruijsenaars-type. The former describe quantum cohomology and quantum K-theory of the ADHM moduli space of Abelian instantons, while the latter arise in the instanton counting of four- and five-dimensional supersymmetric gauge theories with eight supercharges in the presence of defects. Using string theory dualities, we provide correspondences between hydrodynamical and many-body integrable systems. In particular, we match the energy spectra on both sides.     

On Darboux transformations for soliton equations in high-dimensional spacetime

The Darboux transformations for a class of completely integrable systems in the spacetimeR ^{n + 1}, which are much more general than the systems inLett. Math. Phys. 26, 199–209 (1989), are considered. The structure of the nonlinear evolution equations with space constraints is elucidated. It is pointed out that the inverse scattering method can be used to solve the Cauchy problem with initial data given on a noncharacteristic line.Supported by National Basic Research Project Nonlinear Science, NNSFC of China, FEYUT-SEDC-CHINA and Fok Ying-Tung Education Foundation of China.     

Non-Hamiltonian systems separable by Hamilton–Jacobi method

We show that with every separable classical Stäckel system of Benenti type on a Riemannian space one can associate, by a proper deformation of the metric tensor, a multi-parameter family of non-Hamiltonian systems on the same space, sharing the same trajectories and related to the seed system by appropriate reciprocal transformations. These systems are known as bi-cofactor systems and are integrable in quadratures as the seed Hamiltonian system is. We show that with each class of bi-cofactor systems a pair of separation curves can be related. We also investigate the conditions under which a given flat bi-cofactor system can be deformed to a family of geodesically equivalent flat bi-cofactor systems.     

N-dimensional classical integrable systems from Hopf algebras

A systematic method to constructN-body integrable systems is introduced by means of phase space realizations of universal enveloping Hopf algebras. A particular realization for theso(2, 1) case (Gaudin system) is analysed and an integrable quantum deformation is constructed by using quantum algebras as Poisson-Hopf symmetries.Presented at the 5th International Colloquium on Quantum Groups: Quantum Groups and Integrable Systems, Prague, 20–22 June 1996.     

On the kinetic equations of a system of one-dimensional hard rods

The hydrodynamical approximation to an infinite system of one-dimensional identical hard rods interacting through elastic collisions, is shown to be an integrable system possessing a one-parameter family of nonlinear Hamiltonian structures.     

Blow-Up Solutions and Peakons to a Generalized μ-Camassa–Holm Integrable Equation

Considered here is a generalized μ-type integrable equation, which can be regarded as a generalization to both the μ-Camassa–Holm and modified μ-Camassa–Holm equations. It is shown that the proposed equation is formally integrable with the Lax-pair and the bi-Hamiltonian structure and its scale limit is an integrable model of hydrodynamical systems describing short capillary-gravity waves. Local well-posedness of the Cauchy problem in the suitable Sobolev space is established by the viscosity method. Existence of peaked traveling wave solutions and formation of singularities of solutions for the equation are investigated. It is found that the equation admits single and multi-peaked traveling wave solutions. The effects of varying μ-Camassa–Holm and modified μ-Camassa–Holm nonlocal nonlinearities on blow-up criteria and wave breaking are illustrated in detail. Our analysis relies on the method of characteristics and conserved quantities and is proceeded with a priori differential estimates.     

Integrable Rosochatius Deformations of the Restricted cKdV Flows

Three novel finite-dimensional integrable Hamiltonian systems of Rosochatius type and their Lax representations are presented. We make a deformation for the Lax matrbces of the Neumann type, the Bargmann type and the high-order symmetry type of restricted cKdV flows by adding an additional term and then prove that this kind of deformation does not change the r-matrix relations. Finally the new integrable systems are generated from these deformed Lax matrices.     

利用Miura型不可逆变换得到高维可积模型

寻找高维可积模型(特别是3+1维可积模型)是非线性物理中的一个非常重要的问题.建立了一 种利用不可逆形变关系系统寻求高维可积模型的方法.不可逆形变既可以使可积模型成为不 可积模型,也可以使不可积模型成为可积模型.利用一种不可逆的Miura型形变关系和线性波 动方程,得到了一个非平庸的Painlevé可积的高维非线性模型. 关键词： 高维可织模型 不可逆形变 波动方程 Miura型变换     

Theory of tensor invariants of integrable Hamiltonian systems. I. Incompatible Poisson structures

This paper develops a new theory of tensor invariants of a completely integrable non-degenerate Hamiltonian system on a smooth manifoldM ⁿ. The central objects in this theory are supplementary invariant Poisson structuresP _c which are incompatable with the original Poisson structureP ₁ for this Hamiltonian system. A complete classification of invariant Poisson structures is derived in a neighbourhood of an invariant toroidal domain. This classification resolves the well-known Inverse Problem that was brought into prominence by Magri's 1978 paper deveoted to the theory of compatible Poisson structures. Applications connected with the KAM theory, with the Kepler problem, with the basic integrable problem of celestial mechanics, and with the harmonic oscillator are pointed out. A cohomology is defined for dynamical systems on smooth manifolds. The physically motivated concepts of dynamical compatibility and strong dynamical compatibility of pairs of Poisson structures are introduced to study the diversity of pairs of Poisson structures incompatible in Magri's sense. It is proved that if a dynamical systemV preserves two strongly dynamically compatible Poisson structuresP ₁ andP ₂ in a general position then this system is completely integrable. Such a systemV generates a hierarchy of integrable dynamical systems which in general are not Hamiltonian neither with respect toP ₁ nor with respect toP ₂. Necessary conditions for dynamical compatibility and for strong dynamical compatibility are derived which connect these global properties with new local invariants of an arbitrary pair of incompatible Poisson structures.Supported by NSERC grant OGPIN 337.     

On a Class of Integrable Systems with a Cubic First Integral

A few years ago Selivanova gave an existence proof for some integrable models, in fact geodesic flows on two dimensional manifolds, with a cubic first integral. However the explicit form of these models hinged on the solution of a nonlinear third order ordinary differential equation which could not be obtained. We show that an appropriate choice of coordinates allows for integration and gives the explicit local form for the full family of integrable systems. The relevant metrics are described by a finite number of parameters and lead to a large class of models mainly on the manifolds \mathbb S²{{\mathbb S}^2} and \mathbb H²{{\mathbb H}^2} . Many of these systems are globally defined and contain as special cases integrable systems due to Goryachev, Chaplygin, Dullin, Matveev and Tsiganov.     

Hyperelliptic Prym Varieties and Integrable Systems

We introduce two algebraic completely integrable analogues of the Mumford systems which we call hyperelliptic Prym systems, because every hyperelliptic Prym variety appears as a fiber of their momentum map. As an application we show that the general fiber of the momentum map of the periodic Volterra lattice

Modular Classes of Poisson–Nijenhuis Lie Algebroids

The modular vector field of a Poisson–Nijenhuis Lie algebroid A is defined and we prove that, in case of non-degeneracy, this vector field defines a hierarchy of bi-Hamiltonian A-vector fields. This hierarchy covers an integrable hierarchy on the base manifold, which may not have a Poisson–Nijenhuis structure.      

The supersymmetric soliton correlation matrix

Supersymmetric systems in (2/2) dimensions integrable by the supersymmetric generalization of the Zakharov-Shabat ?dressing? method are studied. The supersymmetric version of the ?soliton correlation matrix? is used to obtain multi- soliton solutions to generic supersymmetric systems of Zakharov-Mikhailov- Shabat type, together with their reductions under finite automorphism groups. The sypersymmetric S² sigma model is worked out as an explicit application of the method.     

On the parametrization of finite-gap solutions by frequency and wavenumber vectors and a theorem of I. krichever

We discuss the parametrization of real finite-gap solutions of an integrable equation by frequency and wavenumber vectors. This parametrization underlies perturbation and averaging theories for the finite-gap solutions. Out of the framework of integrable equations, the parametrization gives a convenient coordinate system on the corresponding manifold of Riemann curves.     

The homogeneous Heisenberg subalgebra and equations of AKNS type

To each classicalr-matrix in the finite-dimensional Lie algebrasl(2, ), we associate an integrable hierarchy of evolution equations, a two-dimensional lattice preserved by these flows, and a collection of common conservation laws in terms of a Heisenberg algebra. The different hierarchies related to distinctr-matrices are mapped into one given by means of generalized Miura transformations.Partially supported by the Comisión Asesora de Investigación Científica y Técnica (No. Proyecto PB85-0037).     

Universal Characters and <Emphasis Type="Italic">q</Emphasis>-Painlevé Systems

We propose an integrable system of q-difference equations satisfied by the universal characters and regard it as a q-analogue of the UC hierarchy; see [10]. Via a similarity reduction of this integrable system, rational solutions of the q-Painlevé systems are constructed in terms of the universal characters.     

A Class of Integrable Spin Calogero-Moser Systems

 We introduce a class of spin Calogero-Moser systems associated with root systems of simple Lie algebras and give the associated Lax representations (with spectral parameter) and fundamental Poisson bracket relations. The associated integrable models (called integrable spin Calogero-Moser systems in the paper) and their Lax pairs are then obtained via Poisson reduction and gauge transformations. For Lie algebras of A _n -type, this new class of integrable systems includes the usual Calogero-Moser systems as subsystems. Our method is guided by a general framework which we develop here using dynamical Lie algebroids. Received: 19 October 2001 / Accepted: 7 June 2002 Published online: 21 October 2002 RID="*" ID="*" Research partially supported by NSF grant DMS00-72171.     

Classical foundations of quantum groups

The concept of classical r matrices is developed from a purely canonical standpoint. The final purpose of this work is to bring about a synthesis between recent developments in the theory of integrable systems and the general theory of quantization as a deformation of classical mechanics. The concept of quantization algebra is here dominant; in integrable systems this is the set of dynamical variables that appear in the Lax pair. The nature of this algebra, a solvable Lie algebra in such models as the Sine-Gordon and Toda field theories but semisimple in the case of spin systems, provides a useful scheme for the classification of integrable models. A completely different classification is obtained by the nature of the r matrix employed; there are three kinds: rational, trigonometric, and elliptic. All cases are studied in detail, with numerous examples. Some of the problems connected with quantization are discussed.This paper is dedicated to my friend Asim Barut.     

Quantum Integrable Toda-Like Systems in Deformation Quantization

We define and discuss the notion of quantum integrability of a classically integrable system within the framework of deformation quantization, i.e. the question whether the classical conserved quantities (which are already in involution with respect to the Poisson bracket) commute with respect to some star product on the phase space after possible quantum corrections. As an example of this method, we show by means of suitable 2 by 2 quantum R-matrices that a list of Toda-like classical integrable systems given by Y. B. Suris is quantum integrable with respect to the usual star product of the Weyl type in flat 2n-dimensional space.     

Remarks on the integrable systems associated with rank 2 perturbations

In a paper by Moser, a class of completely integrable systems associated with the rank 2 perturbations of a symmetrical matrixA is given in the case that all eigenvalues ofA are distinct. This problem is also discussed by Alder and van Moerbeke in terms of the Kac-Moody algebra. In this Letter, we prove that these systems are also completely integrable in the case thatA has multiple eigenvalues by use of the moment map and the isospectral deformations.