Relation between the Negative-Order Harry Dym Hierarchy and a Family of Backward Neumann Type Systems

2016 ◽  
Vol 85 (3) ◽  
pp. 034004 ◽  
Author(s):  
Jinbing Chen
2018 ◽  
Vol 15 (03) ◽  
pp. 1850040 ◽  
Author(s):  
Jinbing Chen

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.


2019 ◽  
Vol 32 (03) ◽  
pp. 2050007 ◽  
Author(s):  
Jinbing Chen

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.


2016 ◽  
Vol 30 (32n33) ◽  
pp. 1650396
Author(s):  
Jinbing Chen

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.


2007 ◽  
Vol 7 (1) ◽  
pp. 68-82
Author(s):  
K. Kropielnicka

AbstractA general class of implicit difference methods for nonlinear parabolic functional differential equations with initial boundary conditions of the Neumann type is constructed. Convergence results are proved by means of consistency and stability arguments. It is assumed that given functions satisfy nonlinear estimates of Perron type with respect to functional variables. Differential equations with deviated variables and differential integral problems can be obtained from a general model by specializing given operators. The results are illustrated by numerical examples.


Author(s):  
V. V. Makarov ◽  
D. A. Lozovoy

  Enzootic bovine leucosis (EBL) has been known for more than a century and a half. Its occurrence and registration may have historically been associated with intensive breeding of dairy cattle in Western Europe to increase target productivity. It is known that any limiting intervention in the nature of the animal organism is always accompanied by an uncontrolled and unpredictable change in the genotype of a wider range than the required, particularly negative order. In particular, a decrease in the resistance to macroorganisms and the possibility of the new diseases emergence, including infectious ones (for example, immunodeficiencies such as BLAD syndrome of black-motley cattle and stress syndrome in pigs, the occurrence of scrapie and other slow sheep infections). In the last two decades of the last century, in many disadvantaged countries, primarily Western European, national programs for the eradication of EBL have been developed and subsequently successfully implemented. First of all the motivation was the economy of dairy cattle breeding (mainly the extension of productive age, as well as the tightening of requirements in international trade in cattle and bull products, breeding, pricing, etc.). In an analytical article are reviewed the elements of epizootology of EBL in the foreign countries with special attention to the situation in the USA, scenarios of various control programs, and promising methods for assessing the role of infected animals in the epizootic process. A critical assessment of the problem of EBL in the Russian Federation is given, the reasons for the ineffectiveness of against leucosis measures are discussed.


Sign in / Sign up

Export Citation Format

Share Document