Integrable systems,symmetries, and quantization

These notes are an expanded version of a mini-course given at the Poisson 2016 conference in Geneva. Starting from classical integrable systems in the sense of Liouville, we explore the notion of non-degenerate singularities and expose recent research in connection with semi-toric systems. The quantum and semiclassical counterparts are also presented, in the viewpoint of the inverse question: from the quantum mechanical spectrum, can one recover the classical system?     

Liouville Integrable System and Associated Integrable Coupling

Liouville integrable discrete integrable system is derived based on discrete isospectral problem. It is shown that the hierarchy is completely integrable in the Liouville sense and possesses bi-Hamiltonian structure. Finally, integrable couplings of the obtained system is given by means of semi-direct sums of Lie algebras.     

Liouville Integrable System and Associated Integrable Coupling

Liouville integrable discrete integrable system is derived based on discrete isospectral problem. It is shown that the hierarchy is completely integrable in the Liouville sense and possesses bi-Hamiltonian structure. Finally, integrablecouplings of the obtained system is given by means of semi-direct sums of Lie algebras.     

A Hierarchy of Lax Integrable Lattice Equations,Liouville Integrability and a New Integrable Symplectic Map

A discrete matrix spectral problem and the associated hierarchy of Lax integrable lattice equations are presented, and it is shown that the resulting Lax integrable lattice equations are all Liouville integrable discrete Hamiltonian systems. A new integrable symplectic map is given by binary Bargmann constraint of the resulting hierarchy. Finally, an infinite set of conservation laws is given for the resulting hierarchy.     

Integrable nonlinear equations and Liouville's theorem,II

A symplectic structure is constructed and the Liouville integration carried out for a stationary Lax equation [L, P]=0, whereL is a scalar differential operator of an arbitrary order.n ^th order operators are included into the variety of first-order matrix operators, and properties of this inclusion are studied.     

Integrable nonlinear equations and Liouville's theorem,I

A symplectic structure for stationary Lax equations of the type [L, P]=0 is constructed, whereL is a matrix differential operator of the first order. It is shown that the equation has a sufficient for the complete integrability amount of first integrals in involution. The well-known linearization of the equation by the Abelian mapping is obtained in a natural manner in consequent exercising of Liouville's procedure.     

Discrete Integrable Systems, Positivity, and Continued Fraction Rearrangements

In this review article, we present a unified approach to solving discrete, integrable, possibly non-commutative, dynamical systems, including the Q- and T -systems based on A _r . The initial data of the systems are seen as cluster variables in a suitable cluster algebra, and may evolve by local mutations. We show that the solutions are always expressed as Laurent polynomials of the initial data with non-negative integer coefficients. This is done by reformulating the mutations of initial data as local rearrangements of continued fractions generating some particular solutions, that preserve manifest positivity. We also show how these techniques apply as well to non-commutative settings.     

Integrable optical geometry

It is shown that a Lorentzian 4-manifold admitting a congruence of optical (null) geodesics without shear and twist defines an optical geometry which is integrable (locally flat) in the sense of the theory of G-structures. The existence of a symmetric linear connection compatible with the optical geometry is another condition equivalent to the integrability of the optical G-structure.     

Integrable cosmological potentials

The problem of classification of the Einstein–Friedman cosmological Hamiltonians H with a single scalar inflaton field \(\varphi \), which possess an additional integral of motion polynomial in momenta on the shell of the Friedman constraint \(H=0\), is considered. Necessary and sufficient conditions for the existence of the first-, second- and third-degree integrals are derived. These conditions have the form of ODEs for the cosmological potential \(V(\varphi )\). In the case of linear and quadratic integrals we find general solutions of the ODEs and construct the corresponding integrals explicitly. A new wide class of Hamiltonians that possess a cubic integral is derived. The corresponding potentials are represented in parametric form in terms of the associated Legendre functions. Six families of special elementary solutions are described, and sporadic superintegrable cases are discussed.     

A New Integrable Lattice Hierarchy and Its Two Discrete Integrable Couplings

A new and efficient way is presented for discrete integrable couplings with the help of two semi-direct sum Lie algebras. As its applications, two discrete integrable couplings associated with the lattice equation are worked out. The approach can be used to study other discrete integrable couplings of the discrete hierarchies of solition equations.     

Integrable Rosochatius Deformations for an Integrable Couplings of CKdV Equation Hierarchy

We propose a method to construct the integrable Rosochatius deformations for an integrable couplings equations hierarchy. As applications, the integrable Rosochatius deformations of the coupled CKdV hierarchy with self-consistent sources and its Lax representation are presented.     

Dark Parameterizations,Equivalent Partner Fields and Integrable Systems

After introducing dark parameters into the traditional physical models, some types of new phenomena may be found. An important difficult problem is how to directly observe this kind of physical phenomena. An alternative treatment is to introduce equivalent multiple partner fields. If use this ideal to integrable systems, one may obtain infinitely many new coupled integrable systemsconstituted by the original usual field and partner fields. The idea is illustrated via the celebrate KdV equation. From the procedure, some byproducts can be obtained: A new method to find exact solutions of some types of coupled nonlinear physical problems, say, the perturbation KdV systems, is provided; Some new localized modes such as the staggered modes can be found and some new interaction phenomena like the ghost interaction are discovered.     

Moyal Deformation,Seiberg–Witten Maps,and Integrable Models

A covariant formalism for Moyal deformations of gauge theory and differential equations which determine Seiberg–Witten maps is presented. Replacing the ordinary product of functions by the noncommutative Moyal product, noncommutative versions of integrable models can be constructed. We explore how a Seiberg–Witten map acts in such a framework. As a specific example, we consider a noncommutative extension of the principal chiral model.     

A Hierarchy of Integrable Nonlinear Lattice Equations and New Integrable Symplectic Map

A discrete spectral problem is discussed, and a hierarchy of integrable nonlinear lattice equations related tothis spectral problem is devised. The new integrable symplectic map and finite-dimensional integrable systems are givenby nonlinearization method. The binary Bargmann constraint gives rise to a Backlund transformation for the resultingintegrable lattice equations.     

A Hierarchy of Integrable Lattice Soliton Equations and New Integrable Symplectic Map

Starting from a discrete spectral problem, a hierarchy of integrable lattice soliton equations is derived. It is shown that the hierarchy is completely integrable in the Liouville sense and possesses discrete bi-Hamiltonian structure. A new integrable symplectic map and finite-dimensional integrable systems are given by nonlinearization method. The binary Bargmann constraint gives rise to a Bäcklund transformation for the resulting integrable lattice equations. At last, conservation laws of the hierarchy are presented.     

Generalized Operator Yang-Baxter Equations,Integrable ODEs and Nonassociative Algebras

Abstract

Reductions for systems of ODEs integrable via the standard factorization method (the Adler-Kostant-Symes scheme) or the generalized factorization method, developed by the authors earlier, are considered. Relationships between such reductions, operator Yang-Baxter equations, and some kinds of non-associative algebras are established.