Deformations of loop algebras and integrable systems: hierarchies of integrable equations

2004 ◽  
Vol 45 (12) ◽  
pp. 4578-4595 ◽  
Author(s):  
T. Skrypnyk
Keyword(s):  
Integrable Systems ◽  
Integrable Equations ◽  
Loop Algebras

Lecture on SUSY Gauge Theories and Integrable Systems

1997 ◽  
Vol 11 (26n27) ◽  
pp. 3093-3124
Author(s):  
A. Marshakov
Keyword(s):  
Field Theory ◽  
Quantum Field Theory ◽  
Integrable Systems ◽  
Gauge Theories ◽  
Toda Chain ◽  
Quantum Field ◽  
Standard Quantum ◽  
Integrable Equations ◽  
Complex Curves ◽  
Periodic Toda Chain

I consider main features of the formulation of the finite-gap solutions to integrable equations in terms of complex curves and generating 1-differential. The example of periodic Toda chain solutions is considered in detail. Recently found exact nonperturbative solutions to [Formula: see text] SUSY gauge theories are formulated using the methods of the theory of integrable systems and where possible the parallels between standard quantum field theory results and solutions to the integrable systems are discussed.

DEFORMATIONS OF LOOP ALGEBRAS AND CLASSICAL INTEGRABLE SYSTEMS: FINITE-DIMENSIONAL HAMILTONIAN SYSTEMS

2004 ◽  
Vol 16 (07) ◽  
pp. 823-849 ◽  
Author(s):  
T. SKRYPNYK
Keyword(s):  
Lie Algebras ◽  
Hamiltonian Systems ◽  
Spectral Parameter ◽  
Integrable Systems ◽  
Integrable Hamiltonian Systems ◽  
Infinite Dimensional ◽  
Loop Algebras ◽  
Finite Dimensional ◽  
Quasigraded Lie Algebras ◽  
Classical Integrable

We construct a family of infinite-dimensional quasigraded Lie algebras, that could be viewed as deformation of the graded loop algebras and admit Kostant–Adler scheme. Using them we obtain new integrable hamiltonian systems admitting Lax-type representations with the spectral parameter.

TWO ENLARGED LOOP ALGEBRAS FOR OBTAINING NEW INTEGRABLE HIERARCHIES

2011 ◽  
Vol 25 (19) ◽  
pp. 2637-2656
Author(s):  
YUFENG ZHANG ◽  
HONWAH TAM ◽  
WEI JIANG
Keyword(s):  
Integrable Systems ◽  
Integrable Hierarchies ◽  
Soliton Solutions ◽  
Loop Algebra ◽  
Darboux Transformations ◽  
Soliton Equation ◽  
Hamiltonian Structures ◽  
Soliton Theory ◽  
Loop Algebras ◽  
Soliton Hierarchy

Taking a loop algebra [Formula: see text] we obtain an integrable soliton hierarchy which is similar to the well-known Kaup–Newell (KN) hierarchy, but it is not. We call it a modified KN (mKN) hierarchy. Then two new enlarged loop algebras of the loop algebra [Formula: see text] are established, respectively, which are used to establish isospectral problems. Thus, two various types of integrable soliton-equation hierarchies along with multi-component potential functions are obtained. Their Hamiltonian structures are also obtained by the variational identity. The second hierarchy is integrable couplings of the mKN hierarchy. This paper provides a clue for generating loop algebras, specially, gives an approach for producing new integrable systems. If we obtain a new soliton hierarchy, we could deduce its symmetries, conserved laws, Darboux transformations, soliton solutions and so on. Hence, the way presented in the paper is an important aspect to obtain new integrable systems in soliton theory.

Multi-component matrix loop algebras and their applications to integrable systems

2010 ◽  
Author(s):  
Zhu Li ◽  
Wen Xiu Ma ◽  
Xing-biao Hu ◽  
Qingping Liu
Keyword(s):  
Integrable Systems ◽  
Loop Algebras ◽  
Component Matrix

scholarly journals EXACT SOLUTIONS TO QUANTUM FIELD THEORIES AND INTEGRABLE EQUATIONS

1996 ◽  
Vol 11 (14) ◽  
pp. 1169-1183 ◽  
Author(s):  
A. MARSHAKOV
Keyword(s):  
Exact Solutions ◽  
Gauge Field ◽  
Integrable Systems ◽  
Gauge Field Theories ◽  
Field Theories ◽  
Quantum Field Theories ◽  
Quantum Field ◽  
Integrable Equations ◽  
First Order

The exact solutions to quantum string and gauge field theories are discussed and their formulation in the framework of integrable systems is presented. In particular we consider in detail several examples of the appearance of solutions to the first-order integrable equations of hydrodynamical type and stress that all known examples can be treated as partial solutions to the same problem in the theory of integrable systems.

scholarly journals Contractions of Integrable Equations

1997 ◽  
Vol 12 (01) ◽  
pp. 183-188 ◽  
Author(s):  
N. A. Gromov ◽  
I. V. Kostyakov ◽  
V. V. Kuratov
Keyword(s):  
Integrable Systems ◽  
Initial System ◽  
Integrable Equations

The contraction is applied to obtaining of integrable systems associated with nonsemisimple algebras. The effect of contraction is splitting off some components from initial system without loss of integrability.

scholarly journals Nonperturbative Quantum Theories and Integrable Equations

1997 ◽  
Vol 12 (09) ◽  
pp. 1607-1649 ◽  
Author(s):  
A. Marshakov
Keyword(s):  
Integrable Systems ◽  
Gauge Theories ◽  
Exact Results ◽  
Quantum Theories ◽  
Integrable Equations ◽  
Spectral Curves ◽  
Associativity Equations ◽  
Classical Integrable ◽  
String Theories

I review the appearance of classical integrable systems as an effective tool for the description of nonperturbative exact results in quantum string and gauge theories. Various aspects of this relation: spectral curves, action-angle variables, Whitham deformations and associativity equations are considered separately demonstrating hidden parallels between topological 2d string theories and naively nontopological 4d theories. The proofs are supplemented by explicit illustrative examples.

ON THE STUDY OF NONLINEAR INTEGRABLE SYSTEMS IN (2+1) DIMENSIONS BY DRINFELD-SOKOLOV METHOD

1995 ◽  
Vol 10 (37) ◽  
pp. 2843-2852
Author(s):  
I. MUKHOPADHYAY ◽  
A. ROYCHOWDHURY
Keyword(s):  
Dynamical Systems ◽  
Poisson Structure ◽  
Integrable Systems ◽  
Hamiltonian Structure ◽  
Complete Integrability ◽  
Kp Equation ◽  
Third Order ◽  
Integrable Equations ◽  
The Third ◽  
Integrable Dynamical Systems

The Drinfeld-Sokolov formalism is extended to the case of operator-valued affine Lie algebra to derive nonlinear integrable dynamical systems in (2+1) dimensions. The Poisson structure of these integrable equations are also worked out. While from the first- and second-order flows we get some new integrable equations in (2+1) dimensions, the KP equation is seen to result from the third-order flow. Complete integrability of such equations and the existence of the bi-Hamiltonian structure are demonstrated.

INTEGRABLE SYSTEMS AND DOUBLE-LOOP ALGEBRAS IN STRING THEORY

1991 ◽  
Vol 06 (16) ◽  
pp. 1525-1531 ◽  
Author(s):  
A. MOROZOV
Keyword(s):  
String Theory ◽  
Integrable Systems ◽  
Integrable Models ◽  
Lagrangian Approach ◽  
Double Loop ◽  
Loop Algebras ◽  
Quantum Deformations ◽  
Conformal Models

Entire string theory in the formalism of first quantization cannot be exhausted by the theory of conformal models (CFTs). It is hardly enough to add only 2-dimensional integrable systems. However, examination of these systems may be of use for future guesses and generalizations. The first purpose is to give a unified treatment of all conformal and integrable models. Some steps in this direction are described in the context of Lagrangian approach. The main implication is the need to study 2-loop algebras (like those of fields on [Formula: see text] surfaces) and their quantum deformations.

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