Finite-dimensional Integrable Systems Research Articles

A nonlocal finite-dimensional integrable system related to the nonlocal mKdV equation

Предложена иерархия нелокального модифицированного уравнения Кортевега-де Фриза. Путем наложения связей получены нелокальные конечномерные интегрируемые системы со структурой Ли-Пуассона. С помощью преобразования координат нелокальные гамильтоновы системы Ли-Пуассона сводятся к нелокальным каноническим гамильтоновым системам, имеющим стандартную симплектическую структуру. С помощью нелокальных конечномерных интегрируемых систем получены параметрические решения нелокального модифицированного уравнения Кортевега-де Фриза и обобщенного нелокального нелинейного уравнения Шредингера. На основе теории Гамильтона получены координаты типа действие-угол и задачи обращения, связанные с гамильтоновыми системами Ли-Пуассона.

Lagrangian multiforms on coadjoint orbits for finite-dimensional integrable systems

Lagrangian multiforms provide a variational framework to describe integrable hierarchies. The case of Lagrangian 1-forms covers finite-dimensional integrable systems. We use the theory of Lie dialgebras introduced by Semenov-Tian-Shansky to construct a Lagrangian 1-form. Given a Lie dialgebra associated with a Lie algebra g\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathfrak {g}$$\\end{document} and a collection Hk\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$H_k$$\\end{document}, k=1,⋯,N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$k=1,\\dots ,N$$\\end{document}, of invariant functions on g∗\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathfrak {g}^*$$\\end{document}, we give a formula for a Lagrangian multiform describing the commuting flows for Hk\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$H_k$$\\end{document} on a coadjoint orbit in g∗\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathfrak {g}^*$$\\end{document}. We show that the Euler–Lagrange equations for our multiform produce the set of compatible equations in Lax form associated with the underlying r-matrix of the Lie dialgebra. We establish a structural result which relates the closure relation for our multiform to the Poisson involutivity of the Hamiltonians Hk\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$H_k$$\\end{document} and the so-called double zero on the Euler–Lagrange equations. The construction is extended to a general coadjoint orbit by using reduction from the free motion of the cotangent bundle of a Lie group. We illustrate the dialgebra construction of a Lagrangian multiform with the open Toda chain and the rational Gaudin model. The open Toda chain is built using two different Lie dialgebra structures on sl(N+1)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathfrak {sl}(N+1)$$\\end{document}. The first one possesses a non-skew-symmetric r-matrix and falls within the Adler–Kostant–Symes scheme. The second one possesses a skew-symmetric r-matrix. In both cases, the connection with the well-known descriptions of the chain in Flaschka and canonical coordinates is provided.

A Finite-Dimensional Integrable System Related to the Kadometsev–Petviashvili Equation

In this paper, the Kadometsev–Petviashvili equation and the Bargmann system are obtained from a second-order operator spectral problem Lφ=(∂2−v∂−λu)φ=λφx. By means of the Euler–Lagrange equations, a suitable Jacobi–Ostrogradsky coordinate system is established. Using Cao’s method and the associated Bargmann constraint, the Lax pairs of the differential equations are nonlinearized. Then, a new kind of finite-dimensional Hamilton system is generated. Moreover, involutive representations of the solutions of the Kadometsev–Petviashvili equation are derived.

A nonlocal finite-dimensional integrable system related to the nonlocal nonlinear Schrödinger equation hierarchy

Based on the Lenard gradient sequence, a hierarchy of the nonlocal nonlinear Schrödinger (NNLS) equations is obtained. Using the Lax representation, the nonlocal finite-dimensional integrable system with Lie–Poisson structure is presented. Then, under coordinate transformation, the nonlocal finite-dimensional integrable system with Lie–Poisson structure can be expressed as the canonical Hamiltonian system of the standard symplectic structures. Moreover, the parametric representation of the NNLS equation and the nonlocal complex modified Korteweg–de Vries (NcmKdV) equation are constructed. Finally, according to the Hamilton–Jacobi theory, the action–angle variables are built and the inversion problem related to the Lie–Poisson Hamiltonian systems is discussed.

Symplectic and inverse spectral geometry of integrable systems: A glimpse and open problems

We first give a glimpse of finite dimensional classical integrable Hamiltonian systems from the point of view of symplectic geometry and briefly discuss their quantum counterparts, with an emphasis on recent progress on inverse spectral geometry. Then we propose several open problems about the geometry, topology and dynamics of these systems. The problems are largely motivated by the works of a number of authors, including Arnold, Atiyah, Colin de Verdière, Delzant, Duistermaat, Eliasson, Fomenko, Guillemin, Kolmogorov, Kostant, Moser and Sternberg.

Lax equations for relativistic Gaudin models on elliptic curve

We describe the most general GL NM classical elliptic finite-dimensional integrable system, which Lax matrix has n simple poles on elliptic curve. For M = 1 it reproduces the classical inhomogeneous spin chain, for N = 1 it is the Gaudin type (multispin) extension of the spin Ruijsenaars–Schneider model, and for n = 1 the model of M interacting relativistic GL N tops emerges in some particular case. In this way we present a classification for relativistic Gaudin models on GL-bundles over elliptic curve. As a by-product we describe the inhomogeneous Ruijsenaars chain. We show that this model can be considered as a particular case of multispin Ruijsenaars–Schneider model when residues of the Lax matrix are of rank one. An explicit parametrization of the classical spin variables through the canonical variables is obtained for this model. Finally, the most general GL NM model is also described through R-matrices satisfying associative Yang–Baxter equation. This description provides the trigonometric and rational analogues of GL NM models.

Action-angle variables for the Lie-Poisson Hamiltonian systems associated with the three-wave resonant interaction system

<abstract><p>The $ \mathfrak{gl}_3(\mathbb{C}) $ rational Gaudin model governed by $ 3\times 3 $ Lax matrix is applied to study the three-wave resonant interaction system (TWRI) under a constraint between the potentials and the eigenfunctions. And the TWRI system is decomposed so as to be two finite-dimensional Lie-Poisson Hamiltonian systems. Based on the generating functions of conserved integrals, it is shown that the two finite-dimensional Lie-Poisson Hamiltonian systems are completely integrable in the Liouville sense. The action-angle variables associated with non-hyperelliptic spectral curves are computed by Sklyanin's method of separation of variables, and the Jacobi inversion problems related to the resulting finite-dimensional integrable Lie-Poisson Hamiltonian systems and three-wave resonant interaction system are analyzed.</p></abstract>

Extracting dynamical frequencies from invariants of motion in finite-dimensional nonlinear integrable systems.

Integrable dynamical systems play an important role in many areas of science, including accelerator and plasma physics. An integrable dynamical system with n degrees of freedom possesses n nontrivial integrals of motion, and can be solved, in principle, by covering the phase space with one or more charts in which the dynamics can be described using action-angle coordinates. To obtain the frequencies of motion, both the transformation to action-angle coordinates and its inverse must be known in explicit form. However, no general algorithm exists for constructing this transformation explicitly from a set of n known (and generally coupled) integrals of motion. In this paper we describe how one can determine the dynamical frequencies of the motion as functions of these n integrals in the absence of explicitly known action-angle variables, and we provide several examples.

Spin Ruijsenaars–Schneider Models from Reduction

Spin Ruijsenaars–Schneider model introduced by Krichever and Zabrodin is a generalisation of a finite-dimensional integrable system due to Ruijsenaars and Schneider, where particles carry additional internal degrees of freedom. We review how a Hamiltonian structure of the spin model with the hyperbolic potential can be obtained by means of Poisson reduction applied to a specially chosen Poisson manifold.

Poisson Quasi-Nijenhuis Manifolds and the Toda System

The notion of Poisson quasi-Nijenhuis manifold generalizes that of Poisson-Nijenhuis manifold. The relevance of the latter in the theory of completely integrable systems is well established since the birth of the bi-Hamiltonian approach to integrability. In this note, we discuss the relevance of the notion of Poisson quasi-Nijenhuis manifold in the context of finite-dimensional integrable systems. Generically (as we show by an example with $3$ degrees of freedom) the Poisson quasi-Nijenhuis structure is largely too general to ensure Liouville integrability of a system. However, we prove that the closed (or periodic) $n$-particle Toda lattice can be framed in such a geometrical structure, and its well-known integrals of the motion can be obtained as spectral invariants of a "quasi-Nijenhuis recursion operator", that is, a tensor field $N$ of type $(1,1)$ defined on the phase space of the lattice. This example and some of its generalizations are used to understand whether one can define in a reasonable sense a notion of {\em involutive\} Poisson quasi-Nijenhuis manifold. A geometrical link between the open (or non periodic) and the closed Toda systems is also framed in the context of a general scheme connecting Poisson quasi-Nijenhuis and Poisson-Nijenhuis manifolds.

Recent advances in the monodromy theory of integrable Hamiltonian systems

The notion of monodromy was introduced by J.J. Duistermaat as the first obstruction to the existence of global action coordinates in integrable Hamiltonian systems. This invariant was extensively studied since then and was shown to be non-trivial in various concrete examples of finite-dimensional integrable systems. The goal of the present paper is to give a brief overview of monodromy and discuss some of its generalizations. In particular, we will discuss the monodromy around a focus–focus singularity and the notions of quantum, fractional and scattering monodromy. The exposition will be complemented with a number of examples and open problems.

A differential Galois approach to path integrals

We point out the relevance of the differential Galois theory of linear differential equations for the exact semiclassical computations in path integrals in quantum mechanics. The main tool will be a necessary condition for complete integrability of classical Hamiltonian systems obtained by Ramis and myself [Morales-Ruiz and Ramis, Methods Appl. Anal. 8, 33–96 (2001); see also Morales-Ruiz, in Differential Galois Theory and Non-Integrability of Hamiltonian Systems, Modern Birkhäuser Classics (Springer, Basel, 1999)]; if a finite dimensional complex analytical Hamiltonian system is completely integrable with meromorphic first integrals, then the identity component of the Galois group of the variational equation around any integral curve must be abelian. A corollary of this result is that, for finite dimensional integrable Hamiltonian systems, the semiclassical approach is computable in a closed form in the framework of the differential Galois theory. This explains in a very precise way the success of quantum semiclassical computations for integrable Hamiltonian systems.

A Neumann System of the Third Order Differential Operator Associated with the Boussinesq Equation

Finite-dimensional integrable Hamiltonian system, obtained through the nonlinearization of the 3 × 3 spectral problem associated with the Boussinesq equation, is investigated. A generating function method starting from the Lax-Moser matrix is used to give an effective way to prove the involutivity of integrals. Finite-parameter solution of the Boussinesq equation is calculated based on the commutative system of ordinary differential equations with these integrals as Hamiltonians. The problem of the third order differential operator associated with the Boussinesq Neumann system put forward by H. Flaschka in 1983 is solved.

The expansions of spectral function and the corresponding finite dimensional integrable systems

In this paper, taking the Lax pairs of KdV equation as an illustrative example, with the help of a transformation of the spectral problem, another form of spectral function is used to give finite dimensional integrable systems. These integrable systems are generated by two different kinds of polynomial expansion of spectral function on spectral parameter. Further, the root variables of the spectral function are introduced to study the canonical equations of Hamilton. Finally, based on the Hamilton-Jacobi theory, the action-angle variables are built and the soliton solutions of KdV equation are obtained by inversion.

DECOMPOSING A NEW NONLINEAR DIFFERENTIAL-DIFFERENCE SYSTEM UNDER A BARGMANN IMPLICIT SYMMETRY CONSTRAINT

Firstly, a hierarchy of integrable lattice equations and its bi-Hamilt-onian structures are established by applying the discrete trace identity. Secondly, under an implicit Bargmann symmetry constraint, every lattice equation in the nonlinear differential-difference system is decomposed by an completely integrable symplectic map and a finite-dimensional Hamiltonian system. Finally, the spatial part and the temporal part of the Lax pairs and adjoint Lax pairs are all constrained as finite dimensional Liouville integrable Hamiltonian systems.

Open problems, questions and challenges in finite- dimensional integrable systems.

The paper surveys open problems and questions related to different aspects of integrable systems with finitely many degrees of freedom. Many of the open problems were suggested by the participants of the conference 'Finite-dimensional Integrable Systems, FDIS 2017' held at CRM, Barcelona in July 2017.This article is part of the theme issue 'Finite dimensional integrable systems: new trends and methods'.

Symplectic invariants for parabolic orbits and cusp singularities of integrable systems.

We discuss normal forms and symplectic invariants of parabolic orbits and cuspidal tori in integrable Hamiltonian systems with two degrees of freedom. Such singularities appear in many integrable systems in geometry and mathematical physics and can be considered as the simplest example of degenerate singularities. We also suggest some new techniques which apparently can be used for studying symplectic invariants of degenerate singularities of more general type.This article is part of the theme issue 'Finite dimensional integrable systems: new trends and methods'.

Singular fibres of the Gelfand-Cetlin system on 𝔲(n).

In this paper, we show that every singular fibre of the Gelfand-Cetlin system on co-adjoint orbits of unitary groups is a smooth isotropic submanifold which is diffeomorphic to a two-stage quotient of a compact Lie group by free actions of two other compact Lie groups. In many cases, these singular fibres can be shown to be homogeneous spaces or even diffeomorphic to compact Lie groups. We also give a combinatorial formula for computing the dimensions of all singular fibres, and give a detailed description of these singular fibres in many cases, including the so-called (multi-)diamond singularities. These (multi-)diamond singular fibres are degenerate for the Gelfand-Cetlin system, but they are Lagrangian submanifolds diffeomorphic to direct products of special unitary groups and tori. Our methods of study are based on different ideas involving complex ellipsoids, Lie groupoids and also general ideas coming from the theory of singularities of integrable Hamiltonian systems.This article is part of the theme issue 'Finite dimensional integrable systems: new trends and methods'.

Relations in the cohomology ring of the moduli space of flat SO(2n + 1)-connections on a Riemann surface.

We consider the moduli space of flat SO(2n + 1)-connections (up to gauge transformations) on a Riemann surface, with fixed holonomy around a marked point. There are natural line bundles over this moduli space; we construct geometric representatives for the Chern classes of these line bundles, and prove that the ring generated by these Chern classes vanishes below the dimension of the moduli space, generalizing a conjecture of Newstead.This article is part of the theme issue 'Finite dimensional integrable systems: new trends and methods'.

The U ( n ) Gelfand–Zeitlin system as a tropical limit of Ginzburg–Weinstein diffeomorphisms

In this paper, we show that the Ginzburg–Weinstein diffeomorphism of Alekseev & Meinrenken (Alekseev, Meinrenken 2007 J. Differential Geom. 76 , 1–34. (10.4310/jdg/1180135664)) admits a scaling tropical limit on an open dense subset of . The target of the limit map is a product , where is the interior of a cone, T is a torus, and carries an integrable system with natural action-angle coordinates. The pull-back of these coordinates to recovers the Gelfand–Zeitlin integrable system of Guillemin & Sternberg (Guillemin, Sternberg 1983 J. Funct. Anal. 52 , 106–128. (10.1016/0022-1236(83)90092-7)). As a by-product of our proof, we show that the Lagrangian tori of the Flaschka–Ratiu integrable system on the set of upper triangular matrices meet the set of totally positive matrices for sufficiently large action coordinates. This article is part of the theme issue ‘Finite dimensional integrable systems: new trends and methods’.