New hierarchy of integrable system bi-Hamiltonian structure and constrained flows

We have considered the hierarchy of integrable systems associated with the unstable nonlinear Schrodinger equation. The spectral gradient approach and the trace identity are used to derive the bi-Hamiltonian structure of the system. The bi-Hamiltonian property and the square eigenfunctions determined via the spectral gradient approach are then used to construct constrained flows, which is also proved to be derivable from a rational Lax operator. This new Lax operator of the constrained flows is seen to generate the classical r-matrix. Lastly it is also explicitly demonstrated that the different integrals of motion of the constrained flows Poisson commute.     

On maximally superintegrable systems

Locally any completely integrable system is maximally superintegrable system since we have the necessary number of the action-angle variables. The main problem is the construction of the single-valued additional integrals of motion on the whole phase space by using these multi-valued action-angle variables. Some constructions of the additional integrals of motion for the Stäckel systems and for the integrable systems related with two different quadratic r-matrix algebras are discussed. Among these system there are the open Heisenberg magnet and the open Toda lattices associated with the different root systems.     

Higher-Dimensional Integrable Newton Systems with Quadratic Integrals of Motion

Newton systems     , with integrals of motion quadratic in velocities, are considered. We show that if such a system admits two quadratic integrals of motion of the so-called cofactor type , then it has in fact n quadratic integrals of motion and can be embedded into a  (2 n + 1)  -dimensional bi-Hamiltonian system, which under some non-degeneracy assumptions is completely integrable. The majority of these cofactor pair Newton systems are new, but they also include conservative systems with elliptic and parabolic separable potentials, as well as many integrable Newton systems previously derived from soliton equations. We explain the connection between cofactor pair systems and solutions of a certain system of second-order linear PDEs (the fundamental equations ), and use this to recursively construct infinite families of cofactor pair systems.     

Canonical transformations of the extended phase space and integrable systems

We investigate the explicit construction of a canonical transformation of the time variable and the Hamiltonian whereby a given completely integrable system is mapped into another integrable system. The change of time induces a transformation of the equations of motion and of their solutions, the integrals of motion, the methods of separation of variables, the Lax matrices, and the correspondingr-matrices. For several specific families of integrable systems (Toda chains, Holt systems, and Stäckel-type systems), we construct canonical transformations of time in the extended phase space that preserve the integrability property.     

On Discretization of the Euler Top

The application of intersection theory to construction of n-point finite-difference equations associated with classical integrable systems is discussed. As an example, we present a few new discretizations of motion of the Euler top sharing the integrals of motion with the continuous time system and the Poisson bracket up to the integer scaling factor.     

Construction of Separation Variables for Finite-Dimensional Integrable Systems

We consider a class of integrable systems such that solutions of the corresponding Hamilton–Jacobi equation depend on n+m arbitrary parameters and are represented as products of flat curves. The first n parameters are identified with the values of the integrals of motion. The remaining parameters enter the definition of the integrals of motion as arbitrary constants (charges) and can be used to find separation variables. We show that on the coadjoint orbits of Lie groups, the Casimir operators not only generate a family of integrals but also allow constructing separation variables.     

Periodic solutions for Hamiltonian systems under strong constraining forces

We consider periodic solutions of Hamiltonian systems in Euclidean spaces whose motion is constrained to a submanifold M. We prove that under some nondegeneracy assumptions, periodic solutions persist when the constraint is replaced by a strong restoring potential.     

Algebraic integrability: The Adler-Van Moerbeke approach

In this paper, I present an overview of the active area of algebraic completely integrable systems in the sense of Adler and van Moerbeke. These are integrable systems whose trajectories are straight line motions on abelian varieties (complex algebraic tori). We make, via the Kowalewski-Painlevé analysis, a study of the level manifolds of the systems. These manifolds are described explicitly as being affine part of abelian varieties and the flow can be solved by quadrature, that is to say their solutions can be expressed in terms of abelian integrals. The Adler-Van Moerbeke method’s which will be used is devoted to illustrate how to decide about the algebraic completely integrable Hamiltonian systems and it is primarily analytical but heavily inspired by algebraic geometrical methods. I will discuss some interesting and well known examples of algebraic completely integrable systems: a five-dimensional system, the Hénon-Heiles system, the Kowalewski rigid body motion and the geodesic flow on the group SO(n) for a left invariant metric.     

Self-propulsion of a body with rigid surface and variable coefficient of lift in a perfect fluid

We study the system of a 2D rigid body moving in an unbounded volume of incompressible, vortex-free perfect fluid which is at rest at infinity. The body is equipped with a gyrostat and a so-called Flettner rotor. Due to the latter the body is subject to a lifting force (Magnus effect). The rotational velocities of the gyrostat and the rotor are assumed to be known functions of time (control inputs). The equations of motion are presented in the form of the Kirchhoff equations. The integrals of motion are given in the case of piecewise continuous control. Using these integrals we obtain a (reduced) system of first-order differential equations on the configuration space. Then an optimal control problem for several types of the inputs is solved using genetic algorithms.     

Bi-Hamiltonian systems of natural form

We propose a new method for constructing integrable systems of natural form. In this method, integrals of motion are solutions of an overdetermined system of algebraic and partial differential equations obtained from the compatibility condition for Poisson tensors polynomial in the momenta and from the condition that the bi-Lagrangian distribution corresponding to the integrals of motion is invariant under the action of the recursion operator. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 149, No. 2, pp. 161–182, November, 2006.     

Invariant tori for commuting Hamiltonian PDEs

We generalize to some PDEs a theorem by Eliasson and Nekhoroshev on the persistence of invariant tori in Hamiltonian systems with r integrals of motion and n degrees of freedom, r?n. The result we get ensures the persistence of an r-parameter family of r-dimensional invariant tori. The parameters belong to a Cantor-like set. The proof is based on the Lyapunov-Schmidt decomposition and on the standard implicit function theorem. Some of the persistent tori are resonant. We also give an application to the nonlinear wave equation with periodic boundary conditions on a segment and to a system of coupled beam equations. In the first case we construct 2-dimensional tori, while in the second case we construct 3-dimensional tori.     

Controlled differential equations as Young integrals: A simple approach

The theory of rough paths allows one to define controlled differential equations driven by a path which is irregular. The most simple case is the one where the driving path has finite p-variations with 1?p<2, in which case the integrals are interpreted as Young integrals. The prototypal example is given by stochastic differential equations driven by fractional Brownian motion with Hurst index greater than 1/2. Using simple computations, we give the main results regarding this theory - existence, uniqueness, convergence of the Euler scheme, flow property … - which are spread out among several articles.     

The Kowalevskaya exponents and rational integrability of polynomial differential systems

This paper primarily grows from the paper of Llibre and Zhang [J. Llibre, X. Zhang, Polynomial first integrals for quasi-homogeneous polynomial differential systems, Nonlinearity 15 (2002) 1269-1280] with the following essential generalizations: (i) we prove that the link established in the mentioned paper between the Kowalevskaya exponents and the degree of the polynomial first integrals holds not only for (1,…,1)-2 type systems but also for any (s₁,…,sn)-d type systems. (ii) by using different methods, we obtain necessary and sufficient conditions for planar (s₁,s₂)-d systems to have rational first integrals, whereas in the mentioned paper, only (s₁,s₂)-2 type systems and only polynomial integrability are considered.As an application of the methods and the results, we present an illustrative and well studied example to show its non-existence of polynomial first integrals.     

Multiple integrals involving theH-function of two variables

In this paper, we establish two multiple integrals involving the products of generalised hypergeometric function,H-function of one and two variables. The integrals evaluated are very general in nature and generalise the well-known integrals due to Mittal and Gupta, Goyal, Olkha, Dahiya and others. Some interesting integrals involving the products of elementary special functions and orthogonal polynomials have also been obtained as particular cases of the main results.     

From Jacobi problem of separation of variables to theory of quasipotential Newton equations

Our solution to the Jacobi problem of finding separation variables for natural Hamiltonian systems H = ½p ² + V(q) is explained in the first part of this review. It has a form of an effective criterion that for any given potential V(q) tells whether there exist suitable separation coordinates x(q) and how to find these coordinates, so that the Hamilton-Jacobi equation of the transformed Hamiltonian is separable. The main reason for existence of such criterion is the fact that for separable potentials V(q) all integrals of motion depend quadratically on momenta and that all orthogonal separation coordinates stem from the generalized elliptic coordinates. This criterion is directly applicable to the problem of separating multidimensional stationary Schrödinger equation of quantum mechanics. Second part of this work provides a summary of theory of quasipotential, cofactor pair Newton equations $ \ddot q $ = M(q) admitting n quadratic integrals of motion. This theory is a natural generalization of theory of separable potential systems $ \ddot q $ = ??(q). The cofactor pair Newton equations admit a Hamilton-Poisson structure in an extended 2n + 1 dimensional phase space and are integrable by embedding into a Liouville integrable system. Two characterizations of these systems are given: one through a Poisson pencil and another one through a set of Fundamental Equations. For a generic cofactor pair system separation variables have been found and such system have been shown to be equivalent to a Stäckel separable Hamiltonian system. The theory is illustrated by examples of driven and triangular Newton equations.     

Spatial behavior for some non-standard problems in linear thermoelasticity without energy dissipation

In the present paper we consider a prismatic cylinder occupied by an anisotropic and homogeneous compressible linear thermoelastic material within the framework of the linear theory of thermoelasticity without energy dissipation. The cylinder is subject to zero body force and heat supply and zero lateral specific boundary conditions and the motion is induced by a time-dependent displacement and thermal displacement specified pointwise over the base. Further, the motion is constrained such that the displacement, thermal displacement, velocity and temperature variation at points in the cylinder and at a prescribed time are in given proportions to, but not identical with, their respective initial values. It is shown that certain integrals of the solution spatially evolve with respect to the axial variable. Conditions are derived that show the integrals exhibit alternative behavior and in particular for the semi-infinite cylinder that there is either at least exponential growth or at most exponential decay, provided the elasticity tensor is positive definite or strongly elliptic.     

Superintegrable models on Riemannian surfaces of revolution with integrals of any integer degree (I)

We present a family of superintegrable (SI) systems which live on a Riemannian surface of revolution and which exhibit one linear integral and two integrals of any integer degree larger or equal to 2 in the momenta. When this degree is 2, one recovers a metric due to Koenigs.The local structure of these systems is under control of a linear ordinary differential equation of order n which is homogeneous for even integrals and weakly inhomogeneous for odd integrals. The form of the integrals is explicitly given in the so-called “simple” case (see Definition 2). Some globally defined examples are worked out which live either in H² or in R².     

规范变换与有限维可积系统间的变换

本文表明，利用两个特征值问题的规范变换，不仅可以建立和它们相联系的势的约束之间以及相应的有限维Ｈａｍｉｌｔｏｎ系统间的变换关系式，而且可以由一个可积系统的对合守恒积分导出另一个系统的守恒积分