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.

Quasi-periodic solutions to a negative-order integrable system of 2-component KdV equation

2018 ◽  
Vol 15 (03) ◽  
pp. 1850040 ◽  
Author(s):  
Jinbing Chen
Keyword(s):  
Periodic Solutions ◽  
Integrable System ◽  
Kdv Equation ◽  
Type Systems ◽  
Zero Curvature Representation ◽  
Negative Order ◽  
Symmetric Constraint ◽  
Neumann Type ◽  
Novikov Equation ◽  
Jacobi Inversion

In this paper, the backward and forward Neumann type systems are generalized to deduce the quasi-periodic solutions for a negative-order integrable system of 2-component KdV equation. The 2-component negative-order KdV (2-nKdV) equation is depicted as the zero-curvature representation of two spectral problems. It follows from a symmetric constraint that the 2-nKdV equation is reduced to a pair of backward and forward Neumann type systems, where the involutive solutions of Neumann type systems yield the finite parametric solutions of 2-nKdV equation. The negative-order Novikov equation is given to specify a finite-dimensional invariant subspace for the 2-nKdV flow. With a spectral curve given by the Lax matrix, the 2-nKdV flow is linearized on the Jacobi variety of a Riemann surface, which leads to the quasi-periodic solutions of 2-nKdV equation by using the Riemann-Jacobi inversion.

Finite-Dimensional Hamiltonian Systems from Li Spectral Problem by Symmetry Constraints

2007 ◽  
Vol 62 (7-8) ◽  
pp. 399-405
Author(s):  
Lin Luo ◽  
Engui Fan
Keyword(s):  
Spectral Problem ◽  
Hamiltonian Systems ◽  
Hamiltonian Structure ◽  
Lax Pairs ◽  
Curvature Equation ◽  
Finite Dimensional ◽  
Zero Curvature ◽  
Symmetry Constraints

A hierarchy associated with the Li spectral problem is derived with the help of the zero curvature equation. It is shown that the hierarchy possesses bi-Hamiltonian structure and is integrable in the Liouville sense. Moreover, the mono- and binary-nonlinearization theory can be successfully applied in the spectral problem. Under the Bargmann symmetry constraints, Lax pairs and adjoint Lax pairs are nonlineared into finite-dimensional Hamiltonian systems (FDHS) in the Liouville sense. New involutive solutions for the Li hierarchy are obtained.

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.

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.

Noncommutative negative order AKNS equation and its soliton solutions

2018 ◽  
Vol 33 (35) ◽  
pp. 1850209 ◽  
Author(s):  
H. Wajahat A. Riaz ◽  
Mahmood ul Hassan
Keyword(s):  
Nonlinear Evolution Equation ◽  
Nonlinear Evolution ◽  
Infinite Sequence ◽  
Soliton Solutions ◽  
Lax Pair ◽  
Curvature Condition ◽  
Integrable Structure ◽  
Zero Curvature ◽  
Negative Order ◽  
Akns Equation

A noncommutative negative order AKNS (NC-AKNS(-1)) equation is studied. To show the integrability of the system, we present explicitly the underlying integrable structure such as Lax pair, zero-curvature condition, an infinite sequence of conserved densities, Darboux transformation (DT) and quasideterminant soliton solutions. Moreover, the NC-AKNS(-1) equation is compared with its commutative counterpart not only on the level of nonlinear evolution equation but also for the explicit solutions.

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

ZERO CURVATURE CONDITION OF OSp(2/2) AND THE ASSOCIATED SUPERGRAVITY THEORY

1992 ◽  
Vol 07 (18) ◽  
pp. 4293-4311 ◽  
Author(s):  
ASHOK DAS ◽  
WEN-JUI HUANG ◽  
SHIBAJI ROY
Keyword(s):  
Current Algebra ◽  
Hamiltonian Structure ◽  
Operator Product ◽  
Curvature Condition ◽  
Supergravity Theory ◽  
Kdv Equations ◽  
Zero Curvature ◽  
Anomaly Equation ◽  
Graded Lie Algebra ◽  
The One

The N=2 fermionic extensions of the KdV equations are derived from the zero curvature condition associated with the graded Lie algebra of OSp(2/2). These equations lead to two bi-Hamiltonian systems, one of which is supersymmetric. We also derive the one-parameter family of N=2 supersymmetric KdV equations without a bi-Hamiltonian structure in this approach. Following our earlier proposal, we interpret the zero curvature condition as a gauge anomaly equation which brings out the underlying current algebra for the corresponding 2D supergravity theory. This current algebra is then used to obtain the operator product expansions of various fields of this theory.

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