Two kinds of finite-dimensional integrable reduction to the Harry–Dym hierarchy

2016 ◽  
Vol 30 (32n33) ◽  
pp. 1650396
Author(s):  
Jinbing Chen
Keyword(s):  
Hamiltonian Systems ◽  
Tangent Bundle ◽  
Type Systems ◽  
Lax Pair ◽  
Symplectic Space ◽  
Spatial Variables ◽  
Parametric Solutions ◽  
Finite Dimensional ◽  
Neumann Type ◽  
Temporal And Spatial

In this paper, two kinds of finite-dimensional integrable reduction are studied for the Harry–Dym (HD) hierarchy. From the nonlinearization of Lax pair, the HD hierarchy is reduced to a class of finite-dimensional Hamiltonian systems (FDHSs) in view of a Bargmann map and a set of Neumann type systems by a Neumann map, which separate temporal and spatial variables on the symplectic space [Formula: see text] and the tangent bundle of ellipsoid [Formula: see text], respectively. It turns out that involutive solutions of the resulted finite-dimensional integrable systems (FDISs) directly give rise to finite parametric solutions of HD hierarchy through the Bargmann and Neumann maps. The finite-gap potential to the high-order stationary HD equation is obtained that cuts out a finite-dimensional invariant subspace for the HD flows. Finally, some comparisons of two kinds of integrable reductions are then discussed.

Quasi-Periodic Solutions to the Mixed Kaup-Newell Hierarchy

2018 ◽  
Vol 73 (7) ◽  
pp. 579-593
Author(s):  
Jinbing Chen
Keyword(s):  
Riemann Surface ◽  
Periodic Solutions ◽  
Hamiltonian Systems ◽  
Lax Pair ◽  
Spatial Variables ◽  
Jacobi Variety ◽  
Parametric Solutions ◽  
Finite Dimensional ◽  
Temporal And Spatial ◽  
Jacobi Inversion

AbstractThe mixed Kaup-Newell (mKN) hierarchy, including the nonholonomic deformation of the KN equation, is obtained in the Lenard scheme. By the nonlinearisation of the Lax pair, the mKN hierarchy is reduced to a family of mixed, finite-dimensional Hamiltonian systems (FDHSs) that separate its temporal and spatial variables. It turns out that the Bargmann map not only gives rise to the finite parametric solutions of the mKN hierarchy but also specifies a finite-dimensional, invariant subspace for the mKN flows. The Abel-Jacobi variables are selected to linearise the mKN flows on the Jacobi variety of a Riemann surface, from which some quasi-periodic solutions of mKN hierarchy are presented by using the Riemann-Jacobi inversion.

AN INTEGRABLE DECOMPOSITION OF THE SYMMETRIC MATRIX KdV EQUATION

2008 ◽  
Vol 22 (13) ◽  
pp. 1307-1315
Author(s):  
RUGUANG ZHOU ◽  
ZHENYUN QIN
Keyword(s):  
Hamiltonian Systems ◽  
Matrix Method ◽  
Kdv Equation ◽  
Symmetric Matrix ◽  
Lax Pair ◽  
Integrable Hamiltonian Systems ◽  
R Matrix ◽  
Finite Dimensional ◽  
Higher Rank ◽  
Gaudin Models

A technique for nonlinearization of the Lax pair for the scalar soliton equations in (1+1) dimensions is applied to the symmetric matrix KdV equation. As a result, a pair of finite-dimensional integrable Hamiltonian systems, which are of higher rank generalization of the classic Gaudin models, are obtained. The integrability of the systems are shown by the explicit Lax representations and r-matrix method.

scholarly journals Quasi-periodic solutions of the negative-order Jaulent–Miodek hierarchy

2019 ◽  
Vol 32 (03) ◽  
pp. 2050007 ◽  
Author(s):  
Jinbing Chen
Keyword(s):  
Periodic Solutions ◽  
Hamiltonian Structure ◽  
Spectral Curve ◽  
Type Systems ◽  
Finite Dimensional ◽  
Zero Curvature ◽  
Negative Order ◽  
Neumann Type ◽  
Jacobi Inversion ◽  
Negative Formula

A uniform construction of quasi-periodic solutions to the negative-order Jaulent–Miodek (nJM) hierarchy is presented by using a family of backward Neumann type systems. From the backward Lenard gradients, the nJM hierarchy is put into the zero-curvature setting and the bi-Hamiltonian structure displaying its integrability. The nonlinearization of Lax pair is generalized to the nJM hierarchy such that it can be reduced to a sequence of backward Neumann type systems, whose involutive solutions yield finite parametric solutions of the nJM hierarchy. The negative [Formula: see text]-order stationary JM equation is given to specify a finite-dimensional invariant subspace for the nJM flows. With a spectral curve determined by the Lax matrix, the nJM flows are linearized on the Jacobi variety of a Riemann surface. Finally, the Riemann–Jacobi inversion is applied to Abel–Jacobi solutions of the nJM flows, by which some quasi-periodic solutions are obtained for the nJM hierarchy.

scholarly journals Bargmann Type Systems for the Generalization of Toda Lattices

2014 ◽  
Vol 2014 ◽  
pp. 1-8
Author(s):  
Fang Li ◽  
Liping Lu
Keyword(s):  
Hamiltonian Systems ◽  
Toda Lattice ◽  
Type Systems ◽  
Integrals Of Motion ◽  
Lattice Equation ◽  
Finite Dimensional ◽  
Symplectic Map ◽  
Representation Of Solutions ◽  
Toda Lattice Equation ◽  
Toda Lattices

Under a constraint between the potentials and eigenfunctions, the nonlinearization of the Lax pairs associated with the discrete hierarchy of a generalization of the Toda lattice equation is proposed, which leads to a new symplectic map and a class of finite-dimensional Hamiltonian systems. The generating function of the integrals of motion is presented, by which the symplectic map and these finite-dimensional Hamiltonian systems are further proved to be completely integrable in the Liouville sense. Finally, the representation of solutions for a lattice equation in the discrete hierarchy is obtained.

scholarly journals Feedback stabilization of one-dimensional parabolic systems related to formations

2015 ◽  
Vol 63 (1) ◽  
pp. 295-303
Author(s):  
H. Sano
Keyword(s):  
Finite Dimension ◽  
Time Derivative ◽  
Parabolic Systems ◽  
Feedback Stabilization ◽  
Boundary Values ◽  
One Dimensional ◽  
Finite Dimensional ◽  
Neumann Type ◽  
Sturm Liouville ◽  
Output Operator

Abstract This paper is concerned with the problem of stabilizing one-dimensional parabolic systems related to formations by using finitedimensional controllers of a modal type. The parabolic system is described by a Sturm-Liouville operator, and the boundary condition is different from any of Dirichlet type, Neumann type, and Robin type, since it contains the time derivative of boundary values. In this paper, it is shown that the system is formulated as an evolution equation with unbounded output operator in a Hilbert space, and further that it is stabilized by using an RMF (residual mode filter)-based controller which is of finite-dimension. A numerical simulation result is also given to demonstrate the validity of the finite-dimensional controller

scholarly journals Creative Clusters: Towards the Governance of the Creative Industries Production System?

2004 ◽  
Vol 112 (1) ◽  
pp. 50-66 ◽  
Author(s):  
Andy C. Pratt
Keyword(s):  
Production System ◽  
Cluster Model ◽  
Creative Industries ◽  
Spatial Variables ◽  
Individual Firm ◽  
Alternative Approach ◽  
Temporal And Spatial

The aim of this paper is to critically assess the notion of the creative cluster, and to consider whether it is an appropriate tool for the governance of the creative industries, or even a suitable point from which to begin an analysis of the creative industries. The paper argues that creative clusters are formally a subset of business clusters. A critique of the business clusters literature highlights its shortcomings: a focus on individual firm preferences and a lack of attention to non-economic, situated temporal and spatial variables; a lack of attention to the specificity of particular industries and their associated regulatory peculiarities; and finally, information issues associated with the operationalisation of the cluster model. The paper concludes with a discussion of an alternative approach, looking at a creative industries production system that would better meet the concerns of those seeking to govern the creative industries and creative clusters.

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