Finite-dimensional Integrable Hamiltonian Systems Research Articles

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>

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.

Finite-dimensional Liouville integrable Hamiltonian systems generated from Lax pairs of a bi-Hamiltonian soliton hierarchy by symmetry constraints

We construct finite-dimensional Hamiltonian systems by means of symmetry constraints from the Lax pairs and adjoint Lax pairs of a bi-Hamiltonian hierarchy of soliton equations associated with the 3-dimensional special linear Lie algebra, and discuss the Liouville integrability of these systems based on the existence of sufficiently many integrals of motion.

A new integrable symplectic map by the binary nonlinearization to the super AKNS system

Based on the constructed new Lie super-algebra from OSP(2,2), the super bi-Hamiltonian structure of a new super AKNS hierarchy is obtained by making use of super-trace identity. For the new super AKNS system, an explicit symmetry constraint between the potentials and the eigenfunctions is proposed. Moreover, the super AKNS system is decomposed into two compatible finite-dimensional super integrable systems and the obtained super systems are proved to be finite-dimensional super integrable Hamiltonian systems in the super-symmetry manifold R 4 N ∣ 4 N .

Hamiltonian and symplectic symmetries: An introduction

Classical mechanical systems are modeled by a symplectic manifold $(M,\omega)$, and their symmetries, encoded in the action of a Lie group $G$ on $M$ by diffeomorphisms that preserves $\omega$. These actions, which are called symplectic, have been studied in the past forty years, following the works of Atiyah, Delzant, Duistermaat, Guillemin, Heckman, Kostant, Souriau, and Sternberg in the 1970s and 1980s on symplectic actions of compact abelian Lie groups that are, in addition, of type, i.e. they also satisfy Hamilton's equations. Since then a number of connections with combinatorics, finite dimensional integrable Hamiltonian systems, more general symplectic actions, and topology, have flourished. In this paper we review classical and recent results on Hamiltonian and non Hamiltonian symplectic group actions roughly starting from the results of these authors. The paper also serves as a quick introduction to the basics of symplectic geometry.

Two-Component Super AKNS Equations and Their Finite-Dimensional Integrable Super Hamiltonian System

Starting from a matrix Lie superalgebra, two-component super AKNS system is constructed. By making use of monononlinearization technique of Lax pairs, we find that the obtained two-component super AKNS system is a finite-dimensional integrable super Hamiltonian system. And its Lax representation andr-matrix are also given in this paper.

Bargmann Symmetry Constraint and Binary Nonlinearization of Super NLS-MKdV Hierarchy

An explicit Bargmann symmetry constraint is computed and its associated binary nonlinearization of Lax pairs is carried out for the super NLS-MKdV hierarchy. Under the obtained symmetry constraint, the n-th flow of the super NLS-MKdV hierarchy is decomposed into two super finite-dimensional integrable Hamiltonian systems, defined over the super-symmetry manifold R4N｜2N with the corresponding dynamical variables x and tn. The integrals of motion required for Liouville integrability are explicitly given.

First steps in symplectic and spectral theory of integrable systems

The paper intends to lay out the first steps towards constructing a unified framework to understand the symplectic and spectral theory of finite dimensional integrable Hamiltonian systems. While it is difficult to know what the best approach to such a large classification task would be, it is possible to single out some promising directions and preliminary problems. This paper discusses them and hints at a possible path, still loosely defined, to arrive at a classification. It mainly relies on recent progress concerning integrable systems with only non-hyperbolic and non-degenerate singularities.  &nbsp This work originated in an attempt to develop a theory aimed at answering some questions in quantum spectroscopy. Even though quantum integrable systems date back to the early days of quantum mechanics, such as the work of Bohr, Sommerfeld and Einstein, the theory did not blossom at the time. The development of semiclassical analysis with microlocal techniques in the last forty years now permits a constant interplay between spectral theory and symplectic geometry. A main goal of this paper is to emphasize the symplectic issues that are relevant to quantum mechanical integrable systems, and to propose a strategy to solve them.

An Integrable Symplectic Map of a Differential-Difference Hierarchy

By choosing a discrete matrix spectral problem, a hierarchy of integrable differential-difference equations is derived from the discrete zero curvature equation, and the Hamiltonian structures are built. Through a higher-order Bargmann symmetry constraint, the spatial part and the temporal part of the Lax pairs and adjoint Lax pairs, which we obtained are respectively nonlinearized into a new integrable symplectic map and a finite-dimensional integrable Hamiltonian system in Liouville sense.

The quasi-periodic solutions of mixed KdV equations

The quasi-periodic solutions of mixed KdV equations are constructed. To derive the solutions, we introduce the mixed finite-dimensional integrable Hamiltonian systems (FDIHS), which are linearized on the Jacobi variety of the associated hyperelliptic curve. A relation between the FDIHS and the mixed KdV equations is revealed. Based on this relation, we construct the quasi-periodic solutions for the mixed KdV equations with aid of the Riemann theta function. The quasi-periodic solutions for the KdV6 equation and the negative order KdV equation are obtained as special members of the mixed KdV equations.

New Lax pairs of the Toda lattice and the nonlinearization under a higher-order Bargmann constraint

By starting from a new discrete spectral problem, the Toda lattice is derived through the discrete zero curvature equation. Applying the discrete variational identity to the spectral problem will also reach to the bi-Hamiltonian structure of the Toda lattice. Under a higher-order Bargmann symmetry constraint, the new Lax pairs and the adjoint Lax pairs are nonlinearized into integrable symplectic maps and finite-dimensional Liouville integrable Hamiltonian systems. Finally, a Bäcklund transformation of the Toda lattice is obtained.

Finite dimensional Hamiltonian system related to a Lax pair with symplectic and cyclic symmetries

For the 1 + 1 dimensional Lax pair with a symplectic symmetry and cyclic symmetries, it is shown that there is a natural finite-dimensional Hamiltonian system related to it by presenting a unified Lax matrix. The Liouville integrability of the derived finite-dimensional Hamiltonian systems is proved in a unified way. Any solution of these Hamiltonian systems gives a solution of the original PDE. As an application, the two-dimensional hyperbolic Toda equation is considered and the finite-dimensional integrable Hamiltonian system related to it is obtained from the general results.

An integrable decomposition of the Manakov equation

An integrable decomposition of the Manakov equation is presented. A pair of new finite-dimensional integrable Hamiltonian systems which constitute the integrable decomposition of the Manakov equation are obtained. Mathematical subject classification: 37K10.

Binary nonlinearization of the super classical-Boussinesq hierarchy

The symmetry constraint and binary nonlinearization of Lax pairs for the super classical-Boussinesq hierarchy is obtained. Under the obtained symmetry constraint, the n-th flow of the super classical-Boussinesq hierarchy is decomposed into two super finite-dimensional integrable Hamiltonian systems, defined over the super-symmetry manifold with the corresponding dynamical variables x and tn. The integrals of motion required for Liouville integrability are explicitly given.

A novel symmetry constraint of the super cKdV system

A new (1+1)-dimensional integrable system, i.e. the super coupled Korteweg–de Vries (cKdV) system, has been constructed by a super extension of the well-known (1+1)-dimensional cKdV system. For this new system, a novel symmetry constraint between the potential and the eigenfunction can be obtained by means of the binary nonlinearization of its Lax pairs. The constraints for even variables are explicit and the constraints for odd variables are implicit. Under the symmetry constraint, the spacial part and the temporal parts of the equations associated with the Lax pairs for the super cKdV system can be decomposed into the super finite-dimensional integrable Hamiltonian systems on the supersymmetry manifold R4N|2N + 2, whose integrals of motion are explicitly given.

Algebra-geometric constructions of a (2+1)-dimensional discrete integrable model

A new (2+1)-dimensional differential-difference equation is considered. With the aid of the nonlinearization of the Lax pair, the (2+1)-dimensional differential-difference equation is decomposed into a new integrable symplectic map and a class of finite-dimensional integrable Hamiltonian systems. Based on the theory of algebra curve, the continuous flow and discrete flow are straightened out in view of the introduced Abel–Jacobi coordinates.

The Bargmann symmetry constraint and binary nonlinearization of the super Dirac systems

An explicit Bargmann symmetry constraint is computed and its associated binary nonlinearization of Lax pairs is carried out for the super Dirac systems. Under the obtained symmetry constraint, the n-th flow of the super Dirac hierarchy is decomposed into two super finite-dimensional integrable Hamiltonian systems, defined over the super-symmetry manifold R4N|2N with the corresponding dynamical variables x and tn. The integrals of motion required for Liouville integrability are explicitly given.

Decomposition for a 2+1-Dimensional Discrete Integrable Model

A 2+1-dimensional discrete is presented, which is decomposed into a new integrable symplectic map and a class of finite-dimensional integrable Hamiltonian systems, with aid of the nonlinearization of Lax pairs. The system is completely integrable in the Liouville sense.

Integrable Rosochatius deformations of higher-order constrained flows and the soliton hierarchy with self-consistent sources

We propose a systematic method for generalizing the integrable Rosochatius deformations for finite-dimensional integrable Hamiltonian systems to integrable Rosochatius deformations for infinite-dimensional integrable equations. An infinite number of the integrable Rosochatius deformed higher-order constrained flows of some soliton hierarchies, which includes the generalized integrable Hénon–Heiles system, and the integrable Rosochatius deformations of the KdV hierarchy with self-consistent sources, of the AKNS hierarchy with self-consistent sources and of the mKdV hierarchy with self-consistent sources as well as their Lax representations are presented.