Novikov Algebras and a Classification of Multicomponent Camassa–Holm Equations

A class of multicomponent integrable systems associated with Novikov algebras, which interpolate between Korteweg–de Vries (KdV) and Camassa–Holm‐type equations, is obtained. The construction is based on the classification of low‐dimensional Novikov algebras by Bai and Meng. These multicomponent bi‐Hamiltonian systems obtained by this construction may be interpreted as Euler equations on the centrally extended Lie algebras associated with the Novikov algebras. The related bilinear forms generating cocycles of first, second, and third order are classified. Several examples, including known integrable equations, are presented.     

Poisson Difference Schemes for Hamiltonian Systems on Poisson Manifolds

In this paper a systematical method for the construction of Poisson difference schemes with arbitrary order of accuracy for Hamiltonian systems on Poisson manifolds is considered. The transition of such difference schemes from one time-step to the next is a Poisson map. In addition, these schemes preserve all Casimir functions and, under certain conditions, quadratic first integrals of the original Hamiltonian systems. Especially, the arbitrary order centered schemes preserve all Casimir functions and all quadratic first integrals of the original Hamiltonian systems.     

Dirac结构与Dirac流形

引入了Dirac结构的对偶特征对的概念,并给出了相应的可积性条件．利用这些结果,得到在Dirac流形的子流形上自然诱导出Dirac结构的条件,结果改进了Courant T．J．给出的相应条件;还得到Poisson流形在子流形上诱导出Poisson结构的条件,并改进了Weinstein A．和Courant T．J．所给出的相应条件;最后证明了预辛形式的可约Dirac结构与相应商流形上的辛结构之间存在一一对应的关系．     

Integrable equations,related with the Poisson algebra

The general r-matrix scheme for the construction of integrable Hamiltonian systems is applied to a Poisson algebra, i.e., the algebra of functions on a symplectic manifold, the commutator in which is defined by the Poisson bracket. Integrable systems of two types are constructed: Hamiltonian systems on the group of symplectic diffeomorphisms, whose Hamiltonians are sums of a leftinvariant kinetic energy and a potential, and first-order systems of equations for two functions of one variable.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 169, pp. 44–50, 1988.     

一种构造可积Hamilton系统的新方法

建议了一种新的构造可积Hamilton系统的方法。对于给定的Poisson流形,本文利用Dirac-Poisson结构构造其上的新Poisson括号[1],进而获得了新的可积Hamilton系统。构造的Poisson括号一般是非线的,并且这种方法也不同于通常的方法[2～4]。本文还给出了两个实例。     

Derivation of Korteweg–de Vries flow equations from the short‐wave model equation

We perform a multiple‐time scales analysis and compatibility condition to the short‐wave model equation. We derive Korteweg–de Vries flow equation in the bi‐Hamiltonian form as an amplitude equation. Copyright © 2013 John Wiley & Sons, Ltd.     

On Hamiltonian perturbations of hyperbolic systems of conservation laws I: Quasi‐Triviality of bi‐Hamiltonian perturbations

We study the general structure of formal perturbative solutions to the Hamiltonian perturbations of spatially one‐dimensional systems of hyperbolic PDEs v _t + [?( v )]_x = 0. Under certain genericity assumptions it is proved that any bi‐Hamiltonian perturbation can be eliminated in all orders of the perturbative expansion by a change of coordinates on the infinite jet space depending rationally on the derivatives. The main tool is in constructing the so‐called quasi‐Miura transformation of jet coordinates, eliminating an arbitrary deformation of a semisimple bi‐Hamiltonian structure of hydrodynamic type (the quasi‐triviality theorem). We also describe, following [35], the invariants of such bi‐Hamiltonian structures with respect to the group of Miura‐type transformations depending polynomially on the derivatives. © 2005 Wiley Periodicals, Inc.     

Induced and coinduced Banach Lie–Poisson spaces and integrability

The Poisson induction and coinduction procedures are used to construct Banach Lie–Poisson spaces as well as related systems of integrals in involution. This general method applied to the Banach Lie–Poisson space of trace class operators leads to infinite Hamiltonian systems of k-diagonal trace class operators which have infinitely many integrals. The bidiagonal case is investigated in detail.     

An alternative approach to solve the mixed AKNS equations

The algebraic–geometric solutions of the mixed AKNS equations are investigated through a finite-dimensional Lie–Poisson Hamiltonian system, which is generated by the nonlinearization of the adjoint equation related to the AKNS spectral problem. First, each mixed AKNS equation can be decomposed into two compatible Lie–Poisson Hamiltonian flows. Then the separated variables on the coadjoint orbit are introduced to study these Lie–Poisson Hamiltonian systems. Further, based on the Hamilton–Jacobi theory, the relationship between the action-angle coordinates and the Jacobi-inversion problem is established. In the end, using Riemann–Jacobi inversion, the algebraic–geometric solutions of the first three mixed AKNS equations are obtained.     

Generalization of solutions of the Jacobi PDEs associated to time reparametrizations of Poisson systems

The determination of solutions of the Jacobi partial differential equations (PDEs) for finite-dimensional Poisson systems is considered. In particular, a novel procedure for the construction of solution families is developed. Such a procedure is based on the use of time reparametrizations preserving the existence of a Poisson structure. As a result, a method which is valid for arbitrary values of the dimension and the rank of the Poisson structure under consideration is obtained. In this article two main families of time reparametrizations of this kind are characterized. In addition, these results lead to a novel application which is also developed, namely the global and constructive determination of the Darboux canonical form for Poisson systems of arbitrary dimension and rank two, thus improving the local result provided by Darboux' theorem for such a case.     

Duality transformations of Gibbs random fields

The duality of Gibbs fields are examined in three forms: the Poisson summation formula, “electrodynamic” representation, and Hamiltonian duality. New low-temperature expansions are obtained and systems with nonsummable interaction are studied.     

Poisson structures for geometric curve flows in semi-simple homogeneous spaces

We apply the equivariant method of moving frames to investigate the existence of Poisson structures for geometric curve flows in semi-simple homogeneous spaces. We derive explicit compatibility conditions that ensure that a geometric flow induces a Hamiltonian evolution of the associated differential invariants. Our results are illustrated by several examples of geometric interest.     

Lax方程与耦合Poisson流形的动力r-矩阵

对于一个与Poisson流形耦合的动力r-矩阵,我们在相应的Lie双代数胚上构造出一类Lax方程和一族守恒量,希望利用该方法进一步研究可积Hamilton系统.     

Stability of linear almost-Hamiltonian periodic systems

A linear Hamiltonian system with periodic coefficients is subject to a small “dissipative” perturbation that makes it asymptotically stable. The conditions under which the perturbation remains dissipative for all Hamiltonian systems sufficiently close to the original one are discussed.     

Effect of emergent carrying capacity in an eco‐epidemiological system

The paper explores an eco‐epidemiological model of a predator–prey type, where the prey population is subject to infection. The model is basically a combination of S‐I type model and a Rosenzweig–MacArthur predator–prey model. The novelty of this contribution is to consider different competition coefficients within the prey population, which leads to the emergent carrying capacity. We explicitly separate the competition between non‐infected and infected individuals. This emergent carrying capacity is markedly different to the explicit carrying capacities that have been considered in many eco‐epidemiological models. We observed that different intra‐class and inter‐class competition can facilitate the coexistence of susceptible prey‐infected prey–predator, which is impossible for the case of the explicit carrying capacity model. We also show that these findings are closely associated with bi‐stability. The present system undergoes bi‐stability in two different scenarios: (a) bi‐stability between the planner equilibria where susceptible prey co‐exists with predator or infected prey and (b) bi‐stability between co‐existence equilibrium and the planner equilibrium where susceptible prey coexists with infected prey; have been discussed. The conditions for which the system is to be permanent and the global stability of the system around disease‐free equilibrium are worked out. Copyright © 2015 John Wiley & Sons, Ltd.     

Restrictions of Quadratic Forms to Lagrangian Planes,Quadratic Matrix Equations,and Gyroscopic Stabilization

We discuss the symplectic geometry of linear Hamiltonian systems with nondegenerate Hamiltonians. These systems can be reduced to linear second-order differential equations characteristic of linear oscillation theory. This reduction is related to the problem on the signatures of restrictions of quadratic forms to Lagrangian planes. We study vortex symplectic planes invariant with respect to linear Hamiltonian systems. These planes are determined by the solutions of quadratic matrix equations of a special form. New conditions for gyroscopic stabilization are found.     

On M–multisplittings of singular M–matrices with application to Markov chains

Given a singular M–matrix of a linear system, convergent conditions under which iterative schemes based on M–multisplittings are studied. Two of those conditions, the index of the iteration matrix and its spectral radius are investigated and related to those of the M-matrix. Furthermore, a parallel multisplitting iteration scheme for solving singular linear systems is suggested which can be applied to practical problems such as Poisson and elasticity problems under certain boundary conditions, the Neumann problem, and in Markov chains. A discussion of that multisplitting scheme, based on Gauss–Seidel type splittings is given for computing the stationary distribution vector of Markov chains. In this case a computational viable algorithm can be constructed, since only the nonsingularity of one weighting matrix of the multisplitting is needed. © 1998 John Wiley & Sons, Ltd.     

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.