Bi-Hamiltonian structures and singularities of integrable systems

A Hamiltonian system on a Poisson manifold M is called integrable if it possesses sufficiently many commuting first integrals f ₁, … f _s which are functionally independent on M almost everywhere. We study the structure of the singular set K where the differentials df ₁, …, df _s become linearly dependent and show that in the case of bi-Hamiltonian systems this structure is closely related to the properties of the corresponding pencil of compatible Poisson brackets. The main goal of the paper is to illustrate this relationship and to show that the bi-Hamiltonian approach can be extremely effective in the study of singularities of integrable systems, especially in the case of many degrees of freedom when using other methods leads to serious computational problems. Since in many examples the underlying bi-Hamiltonian structure has a natural algebraic interpretation, the technology developed in this paper allows one to reformulate analytic and topological questions related to the dynamics of a given system into pure algebraic language, which leads to simple and natural answers.     

Webs, Lenard schemes, and the local geometry of bi-Hamiltonian Toda and Lax structures

We introduce a criterion that a given bi-Hamiltonian structure admits a local coordinate system where both brackets have constant coefficients. This criterion is applied to the bi-Hamiltonian open Toda lattice in a generic point, which is shown to be locally isomorphic to a Kronecker odd-dimensional pair of brackets with constant coefficients. This shows that the open Toda lattice cannot be locally represented as a product of two bi-Hamiltonian structures. Near, a generic point, the bi-Hamiltonian periodic Toda lattice is shown to be isomorphic to a product of two open Toda lattices (one of which is a (trivial) structure of dimension 1). While the above results might be obtained by more traditional methods, we use an approach based on general results on geometry of webs. This demonstrates the possibility of applying a geometric language to problems on bi-Hamiltonian integrable systems; such a possibility may be no less important than the particular results proved in this paper. Based on these geometric approaches, we conjecture that decompositions similar to the decomposition of the periodic Toda lattice exist in local geometry of the Volterra system, the complete Toda lattice, the multidimensional Euler top, and a regular bi-Hamiltonian Lie coalgebra. We also state general conjectures about the geometry of more general "homogeneous" finite-dimensional bi-Hamiltonian structures. The class of homogeneous structures is shown to coincide with the class of systems integrable by Lenard scheme. The bi-Hamiltonian structures which admit a non-degenerate Lax structure are shown to be locally isomorphic to the open Toda lattice.     

Compatible metrics and the diagonalizability of nonlocally bi-Hamiltonian systems of hydrodynamic type

We study bi-Hamiltonian systems of hydrodynamic type with nonsingular (semisimple) nonlocal bi-Hamiltonian structures. We prove that all such systems of hydrodynamic type are diagonalizable and that the metrics of the bi-Hamiltonian structure completely determine the complete set of Riemann invariants constructed for any such system. Moreover, we prove that for an arbitrary nonsingular (semisimple) nonlocally bi-Hamiltonian system of hydrodynamic type, there exist local coordinates (Riemann invariants) such that all matrix differential-geometric objects related to this system, namely, the matrix (affinor) V_jⁱ(u) of this system of hydrodynamic type, the metrics g ₁ ^ij(u) and g ₂ ^ij(u), the affinor υ_jⁱ(u) = g ₁ ^is(u)g _2,sj(u), and also the affinors (w _1,n)_jⁱ(u) and (w _2,n)_jⁱ(u) of the nonsingular nonlocal bi-Hamiltonian structure of this system, are diagonal in these special “diagonalizing” local coordinates (Riemann invariants of the system). The proof is a natural corollary of the general results of our previously developed theories of compatible metrics and of nonlocal bi-Hamiltonian structures; we briefly review the necessary notions and results in those two theories.     

Some new integrable systems constructed from the bi-Hamiltonian systems with pure differential Hamiltonian operators

When both Hamiltonian operators of a bi-Hamiltonian system are pure differential operators, we show that the generalized Kupershmidt deformation (GKD) developed from the Kupershmidt deformation in [10] offers an useful way to construct new integrable system starting from the bi-Hamiltonian system. We construct some new integrable systems by means of the generalized Kupershmidt deformation in the cases of Harry Dym hierarchy, classical Boussinesq hierarchy and coupled KdV hierarchy. We show that the GKD of Harry Dym equation, GKD of classical Boussinesq equation and GKD of coupled KdV equation are equivalent to the new integrable Rosochatius deformations of these soliton equations with self-consistent sources. We present the Lax pair for these new systems. Therefore the generalized Kupershmidt deformation provides a new way to construct new integrable systems from bi-Hamiltonian systems and also offers a new approach to obtain the Rosochatius deformation of soliton equation with self-consistent sources.     

The Integrability of Dispersive Hunter-Saxton Equation

In this paper, we prove that the dispersive form of Hunter-Saxton equation is a completely integrable and bi-Hamiltonian system.     

Well-posedness,wave breaking and peakons for a modified μ-Camassa–Holm equation

Considered herein is a modified periodic Camassa–Holm equation with cubic nonlinearity which is called the modified μ-Camassa–Holm equation. The proposed equation is shown to be formally integrable with the Lax pair and bi-Hamiltonian structure. Local well-posedness of the initial-value problem to the modified μ-Camassa–Holm equation in the Besov space is established. Existence of peaked traveling-wave solutions and formation of singularities of solutions for the equation are then investigated. It is shown that the equation admits a single peaked soliton and multi-peakon solutions with a similar character of the μ-Camassa–Holm equation. Singularities of the solutions can occur only in the form of wave-breaking, and several wave-breaking mechanisms for solutions with certain initial profiles are described in detail.     

Symplectic realizations and symmetries of a Lotka-Volterra type system

In this paper a Lotka-Volterra type system is considered. For such a system, bi-Hamiltonian formulation, symplectic realizations and symmetries are presented.     

Compatible Nonlocal Poisson Brackets of Hydrodynamic Type and Integrable Hierarchies Related to Them

We construct integrable bi-Hamiltonian hierarchies related to compatible nonlocal Poisson brackets of hydrodynamic type and solve the problem of the canonical form for a pair of compatible nonlocal Poisson brackets of hydrodynamic type. A system of equations describing compatible nonlocal Poisson brackets of hydrodynamic type is derived. This system can be integrated by the inverse scattering problem method. Any solution of this integrable system generates integrable bi-Hamiltonian systems of hydrodynamic type according to explicit formulas. We construct a theory of Poisson brackets of the special Liouville type. This theory plays an important role in the construction of integrable hierarchies.     

Integrability of Kupershmidt Deformations

We prove that the Kupershmidt deformation of a bi-Hamiltonian system is itself bi-Hamiltonian. Moreover, Magri hierarchies of the initial system give rise to Magri hierarchies of Kupershmidt deformations as well. Since Kupershmidt deformations are not written in evolution form, we start with an outline a geometric framework to study Hamiltonian properties of general non-evolution differential equations, developed in Igonin et al. (to appear, 2009) (see also Kersten et al., In: Differential Equations: Geometry, Symmetries and Integrability, Springer, Berlin, 2009).     

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.     

Propagation of singularities for Cauchy problems of semilinear thermoelastic systems with microtemperatures

The propagation of singularities of solutions to the Cauchy problem of a semilinear thermoelastic system with microtemperatures in one space variable is studied. First, by using a diagonalization argument of phase space analysis, the coupled thermoelastic system with microtemperatures will be decoupled microlocally. Second, using a classical bootstrap argument, the property of finite propagation speed of singularities for the semilinear thermoelastic system is obtained. Finally, it is also shown that the microlocal weak singularities propagate along the null bicharacteristics of the hyperbolic operators of the coupled semilinear system (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Remarks on Propagation of Singularities in Thermoelasticity

The propagation of high order weak singularities for the system of homogeneous thermoelasticity in one space variable is studied by using paralinearization and a new decoupling technique introduced by the author (Microlocal analysis in nonlinear thermoelasticity, to appear). For the linear system, one shows that the nonsmooth initial data for the parabolic part lead to singularities in the hyperbolic part of solutions, even when the initial data for that part are identically zero. Both the Cauchy problem and the problem inside of a domain for the semilinear system are considered. It is shown that the propagation of high order singularities is essentially dominated by the hyperbolic operator in the system of thermoelasticity.     

Bi-Hamiltonian Aspects of the Separability of the Neumann System

The Neumann system on the two-dimensional sphere is used as a tool to convey some ideas on the bi-Hamiltonian standpoint on separation of variables. We show that from this standpoint, its separation coordinates and its integrals of motion can be found systematically.     

一类反对称loop代数及其应用

构造了一类3×3的反对称loop代数,由此设计了一个含5个位势函数的等谱问题;利用屠格式导出了一个Liouville可积系统,且拥有双Hamilton结构.     

A new multi-component matrix loop algebra and a multi-component integrable couplings of the NLS-MKdV hierarchy and its Hamiltonian structure

A new multi-component matrix loop algebra is constructed, which is devoted to establishing an isospectral problem. By making use of Tu scheme, the multi-component integrable couplings of the NLS-MKdV hierarchy is obtained, then the bi-Hamiltonian structure of the above system is given.     

On the bi-Hamiltonian structure of Bogoyavlensky system on <Emphasis Type="Italic">so</Emphasis>(4)

We discuss bi-Hamiltonian structure for the Bogoyavlensky system on so(4) with an additional integral of fourth order in momenta. An explicit procedure to find the variables of separation and the separation relations is considered in detail.     

PERIODIC SOLUTIONS OF DELAY DIFFERENTIAL EQUATIONS WITH TWO DELAY VIA BI-HAMILTONIAN SYSTEMS

1 Illtroduction and Statement of the Main ResultIn this paper, we shall study the existence of periodic solutions for the twofOllowing differential delay equationsX'(t) = --f(x(t -- r1))g(x(t -- r2)) -- f(x(t -- r2))g(x(t -- r1)), (1)andX'(t) = f(x(t -- rl))g(x(t -- r2)) f(x(t -- r2))g(x(t -- rl)), (2)where ri (i = l,2) are positive constants. When the function g(x) = 1,equations (1) and (2) become respectivelyIn 1974, Kaplan and Yorke (see [101) proved the existence of periodic so1utions…     

多分量退化的 CH型方程的可积性及其解

研究了M¨untz 有理函数在加权Orlicz 空间内的逼近性质,证明了它在Orlicz 空间内的有界性,利用加权连续模、K-泛函、Hardy-Littlewood 极大函数、H¨older 不等式给出了该有理函数在Orlicz 空间内的加权逼近性质。     

AN UNITED APPROACH TO BOTH THE TM HIERARCHY AND THE COUPLED KdV HIERARCHY

A united model of both the TM hierarchy and the coupled KdV hierarchy is proposed. By using the trace identity, the bi-Hamiltonian structure of the corresponding hierarchy is established. The isospectral problem is nonlinearized as a new completely integrable Hamiltonian system in Liouville sense.     

Reduction and realization in toda and volterra

We construct a new symplectic, bi-Hamiltonian realization of the KM-system by reducing the corresponding one for the Toda lattice. The bi-Hamiltonian pair is constructed using a reduction theorem of Fernandes and Vanhaecke. In this paper we also review the important work of Moser on the Toda and KM-systems.