Explicit solutions of the classical Calogero & Sutherland systems for any root system

Explicit solutions of the classical Calogero (rational with/without harmonic confining potential) and Sutherland (trigonometric potential) systems is obtained by diagonalisation of certain matrices of simple time evolution. The method works for Calogero & Sutherland systems based on any root system. It generalises the well-known results by Olshanetsky and Perelomov for the A type root systems. Explicit solutions of the (rational and trigonometric) higher Hamiltonian flows of the integrable hierarchy can be readily obtained in a similar way for those based on the classical root systems.Comment: 18 pages, LaTeX, no figur

Water resources of the island of Kahoolawe, Hawaii : preliminary findings

Water-Resources Investigations Report 89-420

Volume preserving multidimensional integrable systems and Nambu--Poisson geometry

In this paper we study generalized classes of volume preserving multidimensional integrable systems via Nambu--Poisson mechanics. These integrable systems belong to the same class of dispersionless KP type equation. Hence they bear a close resemblance to the self dual Einstein equation. All these dispersionless KP and dToda type equations can be studied via twistor geometry, by using the method of Gindikin's pencil of two forms. Following this approach we study the twistor construction of our volume preserving systems

DMRG and periodic boundary conditions: a quantum information perspective

We introduce a picture to analyze the density matrix renormalization group (DMRG) numerical method from a quantum information perspective. This leads us to introduce some modifications for problems with periodic boundary conditions in which the results are dramatically improved. The picture also explains some features of the method in terms of entanglement and teleportation.Comment: 4 page

qq-analogue of modified KP hierarchy and its quasi-classical limit

A $q$-analogue of the tau function of the modified KP hierarchy is defined by a change of independent variables. This tau function satisfies a system of bilinear $q$-difference equations. These bilinear equations are translated to the language of wave functions, which turn out to satisfy a system of linear $q$-difference equations. These linear $q$-difference equations are used to formulate the Lax formalism and the description of quasi-classical limit. These results can be generalized to a $q$-analogue of the Toda hierarchy. The results on the $q$-analogue of the Toda hierarchy might have an application to the random partition calculus in gauge theories and topological strings.Comment: latex2e, a4 paper 15 pages, no figure; (v2) a few references are adde

SDiff(2) Toda equation -- hierarchy, τ\tau function, and symmetries

A continuum limit of the Toda lattice field theory, called the SDiff(2) Toda equation, is shown to have a Lax formalism and an infinite hierarchy of higher flows. The Lax formalism is very similar to the case of the self-dual vacuum Einstein equation and its hyper-K\"ahler version, however now based upon a symplectic structure and the group SDiff(2) of area preserving diffeomorphisms on a cylinder $S^1 \times \R$. An analogue of the Toda lattice tau function is introduced. The existence of hidden SDiff(2) symmetries are derived from a Riemann-Hilbert problem in the SDiff(2) group. Symmetries of the tau function turn out to have commutator anomalies, hence give a representation of a central extension of the SDiff(2) algebra.Comment: 16 pages (``vanilla.sty" is attatched to the end of this file after ``\bye" command

Toda Lattice Hierarchy and Generalized String Equations

String equations of the $p$-th generalized Kontsevich model and the compactified $c = 1$ string theory are re-examined in the language of the Toda lattice hierarchy. As opposed to a hypothesis postulated in the literature, the generalized Kontsevich model at $p = -1$ does not coincide with the $c = 1$ string theory at self-dual radius. A broader family of solutions of the Toda lattice hierarchy including these models are constructed, and shown to satisfy generalized string equations. The status of a variety of $c \le 1$ string models is discussed in this new framework.Comment: 35pages, LaTeX Errors are corrected in Eqs. (2.21), (2.36), (2.33), (3.3), (5.10), (6.1), sentences after (3.19) and theorem 5. A few references are update

Dispersionless integrable equations as coisotropic deformations. Extensions and reductions

Interpretation of dispersionless integrable hierarchies as equations of coisotropic deformations for certain algebras and other algebraic structures like Jordan triple systInterpretation of dispersionless integrable hierarchies as equations of coisotropic deformations for certain algebras and other algebraic structures like Jordan triple systems is discussed. Several generalizations are considered. Stationary reductions of the dispersionless integrable equations are shown to be connected with the dynamical systems on the plane completely integrable on a fixed energy level. ems is discussed. Several generalizations are considered. Stationary reductions of the dispersionless integrable equations are shown to be connected with the dynamical systems on the plane completely integrable on a fixed energy level.Comment: 21 pages, misprints correcte

Critical Point of a Symmetric Vertex Model

We study a symmetric vertex model, that allows 10 vertex configurations, by use of the corner transfer matrix renormalization group (CTMRG), a variant of DMRG. The model has a critical point that belongs to the Ising universality class.Comment: 2 pages, 6 figures, short not