The Extended Bigraded Toda hierarchy

We generalize the Toda lattice hierarchy by considering N+M dependent variables. We construct roots and logarithms of the Lax operator which are uniquely defined operators with coefficients that are $\epsilon$-series of differential polynomials in the dependent variables, and we use them to provide a Lax pair definition of the extended bigraded Toda hierarchy. Using R-matrix theory we give the bihamiltonian formulation of this hierarchy and we prove the existence of a tau function for its solutions. Finally we study the dispersionless limit and its connection with a class of Frobenius manifolds on the orbit space of the extended affine Weyl groups of the $A$ series.Comment: 32 pages, corrected typo

Hodge-GUE correspondence and the discrete KdV equation

We prove the conjectural relationship recently proposed in [9] between certain special cubic Hodge integrals of the Gopakumar--Mari\~no--Vafa type [17, 28] and GUE correlators, and the conjecture proposed in [7] that the partition function of these Hodge integrals is a tau function of the discrete KdV hierarchy

On the water-bag model of dispersionless KP hierarchy

We investigate the bi-Hamiltonian structure of the waterbag model of dKP for two component case. One can establish the third-order and first-order Hamiltonian operator associated with the waterbag model. Also, the dispersive corrections are discussed.Comment: 19 page

On bi-Hamiltonian deformations of exact pencils of hydrodynamic type

In this paper we are interested in non trivial bi-Hamiltonian deformations of the Poisson pencil \omega_{\lambda}=\omega_2+\lambda \omega_1=u\delta'(x-y)+\f{1}{2}u_x\delta(x-y)+\lambda\delta'(x-y). Deformations are generated by a sequence of vector fields $\{X_2, X_4,...\}$, where each $X_{2k}$ is homogenous of degree $2k$ with respect to a grading induced by rescaling. Constructing recursively the vector fields $X_{2k}$ one obtains two types of relations involving their unknown coefficients: one set of linear relations and an other one which involves quadratic relations. We prove that the set of linear relations has a geometric meaning: using Miura-quasitriviality the set of linear relations expresses the tangency of the vector fields $X_{2k}$ to the symplectic leaves of $\omega_1$ and this tangency condition is equivalent to the exactness of the pencil $\omega_{\lambda}$. Moreover, extending the results of [17], we construct the non trivial deformations of the Poisson pencil $\omega_{\lambda}$, up to the eighth order in the deformation parameter, showing therefore that deformations are unobstructed and that both Poisson structures are polynomial in the derivatives of $u$ up to that order.Comment: 34 pages, revised version. Proof of Theorem 16 completely rewritten due to an error in the first versio

Topological structure of the many vortices solution in Jackiw-Pi model

We construct an M-solitons solutions in Jackiw-Pi model depends on 5M parameters(two positions, one scale, one phase per solition and one charge of each solution). By using \phi -mapping method, we discuss the topological structure of the self-duality solution in Jackiw-Pi model in terms of gauge potential decomposition. We set up relationship between Chern-Simons vortices solution and topological number which is determined by Hopf indices and and Brouwer degrees. We also give the quantization of flux in this case.Comment: 14 pages, 4 figure

On a Camassa-Holm type equation with two dependent variables

We consider a generalization of the Camassa Holm (CH) equation with two dependent variables, called CH2, introduced by Liu and Zhang. We briefly provide an alternative derivation of it based on the theory of Hamiltonian structures on (the dual of) a Lie Algebra. The Lie Algebra here involved is the same algebra underlying the NLS hierarchy. We study the structural properties of the CH2 hierarchy within the bihamiltonian theory of integrable PDEs, and provide its Lax representation. Then we explicitly discuss how to construct classes of solutions, both of peakon and of algebro-geometrical type. We finally sketch the construction of a class of singular solutions, defined by setting to zero one of the two dependent variables.Comment: 22 pages, 2 figures. A few typos correcte

Involutive orbits of non-Noether symmetry groups

We consider set of functions on Poisson manifold related by continues one-parameter group of transformations. Class of vector fields that produce involutive families of functions is investigated and relationship between these vector fields and non-Noether symmetries of Hamiltonian dynamical systems is outlined. Theory is illustrated with sample models: modified Boussinesq system and Broer-Kaup system.Comment: LaTeX 2e, 10 pages, no figure

On negative flows of the AKNS hierarchy and a class of deformations of bihamiltonian structure of hydrodynamic type

A deformation parameter of a bihamiltonian structure of hydrodynamic type is shown to parameterize different extensions of the AKNS hierarchy to include negative flows. This construction establishes a purely algebraic link between, on the one hand, two realizations of the first negative flow of the AKNS model and, on the other, two-component generalizations of Camassa-Holm and Dym type equations. The two-component generalizations of Camassa-Holm and Dym type equations can be obtained from the negative order Hamiltonians constructed from the Lenard relations recursively applied on the Casimir of the first Poisson bracket of hydrodynamic type. The positive order Hamiltonians, which follow from Lenard scheme applied on the Casimir of the second Poisson bracket of hydrodynamic type, are shown to coincide with the Hamiltonians of the AKNS model. The AKNS Hamiltonians give rise to charges conserved with respect to equations of motion of two-component Camassa-Holm and two-component Dym type equations.Comment: 20 pages, Late

Determination of the D0 -> K+pi- Relative Strong Phase Using Quantum-Correlated Measurements in e+e- -> D0 D0bar at CLEO

We exploit the quantum coherence between pair-produced D0 and D0bar in psi(3770) decays to study charm mixing, which is characterized by the parameters x and y, and to make a first determination of the relative strong phase \delta between doubly Cabibbo-suppressed D0 -> K+pi- and Cabibbo-favored D0bar -> K+pi-. We analyze a sample of 1.0 million D0D0bar pairs from 281 pb^-1 of e+e- collision data collected with the CLEO-c detector at E_cm = 3.77 GeV. By combining CLEO-c measurements with branching fraction input and time-integrated measurements of R_M = (x^2+y^2)/2 and R_{WS} = Gamma(D0 -> K+pi-)/Gamma(D0bar -> K+pi-) from other experiments, we find \cos\delta = 1.03 +0.31-0.17 +- 0.06, where the uncertainties are statistical and systematic, respectively. In addition, by further including external measurements of charm mixing parameters, we obtain an alternate measurement of \cos\delta = 1.10 +- 0.35 +- 0.07, as well as x\sin\delta = (4.4 +2.7-1.8 +- 2.9) x 10^-3 and \delta = 22 +11-12 +9-11 degrees.Comment: 37 pages, also available through http://www.lns.cornell.edu/public/CLNS/2007/. Incorporated referee's comment

Measurement of B(Ds+ -->ell+ nu) and the Decay Constant fDs From 600/pb of e+e- Annihilation Data Near 4170 MeV

We examine e+e- --> Ds^-D_s^{*+} and Ds^{*-}Ds^{+} interactions at 4170 MeV using the CLEO-c detector in order to measure the decay constant fDs with good precision. Previously our measurements were substantially higher than the most precise lattice based QCD calculation of (241 +/- 3) MeV. Here we use the D_s^+ --> ell^+ nu channel, where the ell^+ designates either a mu^+ or a tau^+, when the tau^+ --> pi^+ anti-nu. Analyzing both modes independently, we determine B(D_s^+ --> mu^+ nu)= 0.565 +/- 0.045 +/- 0.017)%, and B(D_s^+ --> mu^+ nu)= (6.42 +/- 0.81 +/- 0.18)%. We also analyze them simultaneously to find an effective value of B^{eff}(D_s^+ --> mu^+ nu)= (0.591 +/- 0.037 +/- 0.018)% and fDs=(263.3 +/- 8.2 +/- 3.9) MeV. Combining with the CLEO-c value determined independently using D_s^+ --> tau^+ nu, tau^+ --> e^+ nu anti-nu decays, we extract fDs=(259.5 +/- 6.6 +/- 3.1) MeV. Combining with our previous determination of B(D^+ --> mu^+ nu), we extract the ratio fDs/fD+=1.26 +/- 0.06 +/- 0.02. No evidence is found for a CP asymmetry between Gamma(D_s^+ --> mu^+\nu) and \Gamma(D_s^- --> mu^- nu); specifically the fractional difference in rates is measured to be (4.8 +/- 6.1)%. Finally, we find B(D_s^+ --> e^+ nu) < 1.2x10^{-4} at 90% confidence level.Comment: 26 pages, 16 figure