Category of nonlinear evolution equations, algebraic structure, and r-matrix

This paper deals with the category of nonlinear evolution equations (NLEEs) associated with the spectral problem and provides an approach for constructing their algebraic structure and $r$-matrix. First we introduce the category of NLEEs, which composes of various positive order and negative order hierarchies of NLEEs both integrable and non-integrable. The whole category of NLEEs possesses a generalized Lax representation. Next, we present two different Lie algebraic structures of the Lax operator, one of them is universal in the category,i.e. independent of the hierarchy, while the other one is nonuniversal in the hierarchy, i.e. dependent on the underlying hierarchy. Moreover, we find that two kinds of adjoint maps are $r$-matrices under the algebraic structures. In particular, the Virasoro algebraic structures without central extension of isospectral and non-isospectral Lax operators can be viewed as reductions of our algebraic structure. Finally, we give several concrete examples to illustrate our methods. Particularly, the Burgers category is linearized when the generator, which generates the category, is chosen to be independent of the potential function. Furthermore, an isospectral negative order hierarchy in the Burger's category is solved with its general solution. Additionally, in the KdV category we find an interesting fact: the Harry-Dym hierarchy is contained in this category as well as the well-known Harry-Dym equation is included in a positive order KdV hierarchy.Comment: 24 pages, 0 figure

Negative order MKdV hierarchy and a new integrable Neumann-like system

The purpose of this paper is to develop the negative order MKdV hierarchy and to present a new related integrable Neumann-like Hamiltonian flow from the view point of inverse recursion operator and constraint method. The whole MKdV hierarchy both positive and negative is generated by the kernel elements of Lenard's operators pair and recursion operator. Through solving a key operator equation, the whole MKdV hierarchy is shown to have the Lax representation. In particular, some new integrable equation together with the Liouville equations, the sine-Gordon equation, and the sinh-Gordon equation are derived from the negative order MKdV hierarchy. It is very interesting that the restricted flow, corresponding to the negative order MKdV hierarchy, is just a new kind of Neumann-like system. This new Neumann-like system is obtained through restricting the MKdV spectral problem onto a symplectic submanifold and is proven to be completely integrable under the Dirac-Poisson bracket, which we define on the symplectic submanifold. Finally, with the help of the constraint between the Neumann-like system and the negative order MKdV hierarchy, all equations in the hierarchy are proven to have the parametric representations of solutions. In particular, we obtain the parametric solutions of the sine-Gordon equation and the sinh-Gordon equation.Comment: 21 pages, 0 figure

Currents and current correlations in a topological superconducting nanowire beam splitter

A beam splitter consisting of two normal leads coupled to one end of a topological superconducting nanowire via double quantum dot is investigated. In this geometry, the linear current cross-correlations at zero temperature change signs versus the overlap between the two Majorana bound states hosted by the nanowire. Under symmetric bias voltages the net current flowing through the nanowire is noiseless. These two features highlight the fermionic nature of such exotic Majorana excitations though they are based on the superconductivity. Moreover, there exists a unique local particle-hole symmetry inherited from the self-Hermitian property of Majorana bound states, which is apparently scarce in other systems. We show that such particular symmetry can be revealed through measuring the currents under complementary bias voltages.Comment: 6 pages, 4 figure

Application analysis on different suture of scleral flap in trabeculectomy

AIM: To research the application of scleral flap suture in trabeculectomy. <p>METHODS: Totally 114 primary angle-closure glaucoma patients, aged from 36-72 years old, were selected as the objects, and randomly divided into research group and control group. The two groups received different administration methods. Traditional sewing method of sclera flap was used in research group and improved sewing method of sclera flap was used in control group. <p>RESULTS: There was statistical differences between postoperative intraocular pressure of the patients in the observation group and the control group after 1d; 2wk; 1, 3mo(<i>P</i><0.05). There was no statistical difference in intraocular pressure between the two groups. There was statistical differences between incidence of shallow anterior chamber of the patients in the observation group and the control group postoperatively early stage(<i>P</i><0.05). After 6mo, the filtering bleb formation in observation group was no significantly better than control group(<i>P</i>>0.05).<p>CONCLUSION: It is safe and effective that the improved sewing method of sclera flap for trabeculectomy of acute angle-closure glaucoma, and it is a better method to avoid the occurrence of shallow anterior chamber than the traditional sewing method in the early stage after operation

Framework for waveband switching in multigranular optical networks: part I-multigranular cross-connect architectures

Optical networks using wavelength-division multiplexing (WDM) are the foremost solution to the ever-increasing traffic in the Internet backbone. Rapid advances in WDM technology will enable each fiber to carry hundreds or even a thousand wavelengths (using dense-WDM, or DWDM, and ultra-DWDM) of traffic. This, coupled with worldwide fiber deployment, will bring about a tremendous increase in the size of the optical cross-connects, i.e., the number of ports of the wavelength switching elements. Waveband switching (WBS), wherein wavelengths are grouped into bands and switched as a single entity, can reduce the cost and control complexity of switching nodes by minimizing the port count. This paper presents a detailed study on recent advances and open research issues in WBS networks. In this study, we investigate in detail the architecture for various WBS cross-connects and compare them in terms of the number of ports and complexity and also in terms of how flexible they are in adjusting to dynamic traffic. We outline various techniques for grouping wavelengths into bands for the purpose of WBS and show how traditional wavelength routing is different from waveband routing and why techniques developed for wavelength-routed networks (WRNs) cannot be simply applied to WBS networks. We also outline how traffic grooming of subwavelength traffic can be done in WBS networks. In part II of this study [Cao , submitted to J. Opt. Netw.], we study the effect of wavelength conversion on the performance of WBS networks with reconfigurable MG-OXCs. We present an algorithm for waveband grouping in wavelength-convertible networks and evaluate its performance. We also investigate issues related to survivability in WBS networks and show how waveband and wavelength conversion can be used to recover from failures in WBS networks