scholarly journals On the spectrum of the Lax operator of the Benjamin-Ono equation on the torus

2020 ◽  
Vol 279 (12) ◽  
pp. 108762
Author(s):  
Patrick Gérard ◽  
Thomas Kappeler ◽  
Petar Topalov
Keyword(s):  
Lax Operator

scholarly journals Classification of central extensions of Lax operator algebras

2008 ◽  
Author(s):  
Martin Schlichenmaier ◽  
Piotr Kielanowski ◽  
Anatol Odzijewicz ◽  
Martin Schlichenmaier ◽  
Theodore Voronov
Keyword(s):  
Operator Algebras ◽  
Central Extensions ◽  
Lax Operator

POISSON STRUCTURE AND INTEGRABILITY OF THE NEUMANN-MOSER-UHLENBECK MODEL

1990 ◽  
Vol 05 (23) ◽  
pp. 4477-4488 ◽  
Author(s):  
J. AVAN ◽  
M. TALON
Keyword(s):  
Poisson Structure ◽  
Point Of View ◽  
Conserved Quantities ◽  
Poisson Brackets ◽  
Quantum Level ◽  
Quantum Integrability ◽  
Commuting Operators ◽  
Lax Operator

Neumann’s model, describing the motion of a particle on an N-sphere under harmonic forces, is studied from the point of view of classical and quantum integrability. Classical integrability is derived from a generalized structure, “R-S couple” or “D-matrix” for the Poisson brackets of the Lax operator. The already-known set of conserved quantities for this model turns out to follow straightforwardly from this structure. It gives rise to a set of commuting operators at the quantum level, and the algebra of Lax operators directly follows from the classical one.

CANONICAL BÄCKLUND TRANSFORMATION — A NEW DISCRETE INTEGRABLE SYSTEM AND BETHE ANSATZ

2005 ◽  
Vol 20 (18) ◽  
pp. 4355-4361
Author(s):  
SUPRIYA MUKHERJEE ◽  
A. ROY CHOWDHURY ◽  
A. GHOSE CHOUDHURY
Keyword(s):  
Explicit Form ◽  
Integrable System ◽  
Canonical Variable ◽  
Bäcklund Transformation ◽  
Backlund Transformation ◽  
R Matrix ◽  
Discrete Integrable System ◽  
Lax Operator ◽  
Bethe Equations ◽  
Quantization Problem

A new discrete Lax operator involving discrete canonical variable is introduced which generate new integrable system, and is analyzed in the light of the new concept of canonical Bäcklund transformation and classical r-matrix. The generating function of the transformation is explicitly deduced. The second half of the paper deals with the quantization problem where an explicit form of the Bethe equations are deduced.

TOPOLOGICAL σ MODELS AND LARGE N MATRIX INTEGRAL

1995 ◽  
Vol 10 (29) ◽  
pp. 4203-4224 ◽  
Author(s):  
TOHRU EGUCHI ◽  
KENTARO HORI ◽  
SUNG-KIL YANG
Keyword(s):  
Matrix Model ◽  
Riemann Surfaces ◽  
Holomorphic Maps ◽  
Integrable Structure ◽  
Toda Hierarchy ◽  
Matrix Integral ◽  
Large N ◽  
The Matrix ◽  
Lax Operator ◽  
The One

In this paper we describe in some detail the representation of the topological CP1 model in terms of a matrix integral which we have introduced in a previous article. We first discuss the integrable structure of the CP1 model and show that it is governed by an extension of the one-dimensional Toda hierarchy. We then introduce a matrix model which reproduces the sum over holomorphic maps from arbitrary Riemann surfaces onto CP1. We compute intersection numbers on the moduli space of curves using a geometrical method and show that the results agree with those predicted by the matrix model. We also develop a Landau-Ginzburg (LG) description of the CP1 model using a superpotential eX + et0,Q e-X given by the Lax operator of the Toda hierarchy (X is the LG field and t0,Q is the coupling constant of the Kähler class). The form of the superpotential indicates the close connection between CP1 and N=2 supersymmetric sine-Gordon theory which was noted sometime ago by several authors. We also discuss possible generalizations of our construction to other manifolds and present an LG formulation of the topological CP2 model.

ALGEBRAIC STUDY ON THE POISSON STRUCTURE OF LAX OPERATOR OF B AND C TYPE AND SUPER LAX OPERATOR.

1992 ◽  
Vol 07 (supp01a) ◽  
pp. 405-418
Author(s):  
KAORU IKEDA
Keyword(s):  
Poisson Structure ◽  
Algebraic Structure ◽  
Lax Equation ◽  
Lax Operator

The Poisson structure of Lax operator of B and C type and super Lax operator which has odd parity are studied. The algebraic structure of Poisson structure as a background of Lax equation is thrown light on.

CANONICAL BÄCKLUND TRANSFORMATION FOR A DISCRETE INTEGRABLE CHAIN AND ITS ASSOCIATED PROPERTIES

2003 ◽  
Vol 18 (16) ◽  
pp. 1127-1139
Author(s):  
A. GHOSE CHOUDHURY ◽  
BARUN KHANRA ◽  
A. ROY CHOWDHURY
Keyword(s):  
Poisson Structure ◽  
Algebraic Structure ◽  
Bäcklund Transformation ◽  
Inverse Transformation ◽  
Strict Sense ◽  
Backlund Transformation ◽  
Lax Equation ◽  
R Matrix ◽  
Lax Operator ◽  
Time Part

The concept of a canonical Bäcklund transformation as laid down by Sklyanin is extended to a discrete integral chain, with a Poisson structure which is not canonical in the strict sense. The transformation is induced by an auxiliary Lax operator with a classical r-matrix which is similar in its algebraic structure to that of the original Lax operator governing the dynamics of the chain. Moreover, the transformation can be obtained from a suitable generating function. It is also shown how successive transformations can be composed to construct a new transformation. Finally an inverse transformation is also constructed. The compatibility of the transformation with the "time" part of the Lax equation is explicitly demonstrated. It is also shown that the Bianchi theorem of permutability holds good.

scholarly journals Weak KAM Solutions of a Discrete-Time Hamilton-Jacobi Equation in a Minimax Framework

2013 ◽  
Vol 2013 ◽  
pp. 1-12
Author(s):  
Porfirio Toledo
Keyword(s):  
Discrete Time ◽  
Jacobi Equation ◽  
Kam Theory ◽  
Infinite Time ◽  
Discount Factors ◽  
Long Time ◽  
Lax Operator ◽  
Discrete Time Case ◽  
Zero Sum ◽  
Hamilton Jacobi Equation

The purpose of this paper is to study the existence of solutions of a Hamilton-Jacobi equation in a minimax discrete-time case and to show different characterizations for a real number called the critical value, which plays a central role in this work. We study the behavior of solutions of this problem using tools of game theory to obtain a “fixed point” of the Lax operator associated, considering some facts of weak KAM theory to interpret these solutions as discrete viscosity solutions. These solutions represent the optimal payoff of a zero-sum game of two players, with increasingly long time payoffs. The developed techniques allow us to study the behavior of an infinite time game without using discount factors or average actions.

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