scholarly journals SELECTED TOPICS IN CLASSICAL INTEGRABILITY

2012 ◽  
Vol 27 (05) ◽  
pp. 1230003 ◽  
Author(s):  
ANASTASIA DOIKOU
Keyword(s):  
Integrable Systems ◽  
Toda Chain ◽  
Lax Pair ◽  
Integrable Models ◽  
Integrals Of Motion ◽  
Curvature Condition ◽  
R Matrix ◽  
Zero Curvature ◽  
Algebraic Description ◽  
Classical Integrable

Basic notions regarding classical integrable systems are reviewed. An algebraic description of the classical integrable models together with the zero curvature condition description is presented. The classical r-matrix approach for discrete and continuum classical integrable models is introduced. Using this framework the associated classical integrals of motion and the corresponding Lax pair are extracted based on algebraic considerations. Our attention is restricted to classical discrete and continuum integrable systems with periodic boundary conditions. Typical examples of discrete (Toda chain, discrete NLS model) and continuum integrable models (NLS, sine–Gordon models and affine Toda field theories) are also discussed.

Noncommutative negative order AKNS equation and its soliton solutions

2018 ◽  
Vol 33 (35) ◽  
pp. 1850209 ◽  
Author(s):  
H. Wajahat A. Riaz ◽  
Mahmood ul Hassan
Keyword(s):  
Nonlinear Evolution Equation ◽  
Nonlinear Evolution ◽  
Infinite Sequence ◽  
Soliton Solutions ◽  
Lax Pair ◽  
Curvature Condition ◽  
Integrable Structure ◽  
Zero Curvature ◽  
Negative Order ◽  
Akns Equation

A noncommutative negative order AKNS (NC-AKNS(-1)) equation is studied. To show the integrability of the system, we present explicitly the underlying integrable structure such as Lax pair, zero-curvature condition, an infinite sequence of conserved densities, Darboux transformation (DT) and quasideterminant soliton solutions. Moreover, the NC-AKNS(-1) equation is compared with its commutative counterpart not only on the level of nonlinear evolution equation but also for the explicit solutions.

scholarly journals Zero Curvature Formalism for Supersymmetric Integrable Hierarchies in Superspace

1997 ◽  
Vol 12 (34) ◽  
pp. 2623-2630 ◽  
Author(s):  
H. Aratyn ◽  
C. Rasinariu ◽  
A. Das
Keyword(s):  
Symmetric Space ◽  
Integrable Systems ◽  
Integrable Hierarchies ◽  
Space Structure ◽  
Hermitian Symmetric Space ◽  
Curvature Condition ◽  
Lax Equation ◽  
Abelian Gauge ◽  
Graded Algebras ◽  
Zero Curvature

We generalize the Drinfeld–Sokolov formalism of bosonic integrable hierarchies to superspace, in a way which systematically leads to the zero curvature formulation for the supersymmetric integrable systems starting from the Lax equation in superspace. We use the method of symmetric space as well as the non-Abelian gauge technique to obtain the supersymmetric integrable hierarchies of the AKNS type from the zero curvature condition in superspace with the graded algebras, sl (n+1,n), providing the Hermitian symmetric space structure.

scholarly journals ON LIE GROUPS, CLUSTER VARIABLES AND INTEGRABLE SYSTEMS

2013 ◽  
Vol 28 (03n04) ◽  
pp. 1340007
Author(s):  
A. MARSHAKOV
Keyword(s):  
Lie Groups ◽  
Integrable Systems ◽  
Explicit Construction ◽  
Integrable Models ◽  
Loop Groups ◽  
Integrals Of Motion ◽  
General Terms

We propose an explicit construction for the integrable models on Poisson submanifolds of the Lie groups. The integrals of motion are computed in cluster variables via the Lax map. This generalized construction for the co-extended loop groups allows to formulate, in general terms, some new classes of integrable models.

scholarly journals A QUANTUM BRST–ANTI-BRST APPROACH TO CLASSICAL INTEGRABLE SYSTEMS

2004 ◽  
Vol 19 (09) ◽  
pp. 693-702 ◽  
Author(s):  
MICHAEL CHESTERMAN ◽  
MARCELO B. SILKA
Keyword(s):  
Wave Function ◽  
Quantum Theory ◽  
Inverse Scattering ◽  
Integrable Systems ◽  
Classical Mechanics ◽  
Lax Pair ◽  
Liouville Integrability ◽  
Lax Equation ◽  
Scattering Wave ◽  
Classical Integrable

We reformulate the conditions of Liouville integrability in the language of Gozzi et al.'s quantum BRST–anti-BRST description of classical mechanics. The Das–Okubo geometrical Lax equation is particularly suited for this approach. We find that the Lax pair and inverse scattering wave function appear naturally in certain sectors of the quantum theory.

ZERO CURVATURE CONDITION AND 2D GRAVITY THEORIES

1992 ◽  
Vol 07 (15) ◽  
pp. 3447-3472 ◽  
Author(s):  
A. DAS ◽  
W.-J. HUANG ◽  
S. ROY
Keyword(s):  
Current Algebra ◽  
2D Gravity ◽  
Integrable Model ◽  
Integrable Models ◽  
Operator Product ◽  
Curvature Condition ◽  
Kdv Hierarchy ◽  
Zero Curvature ◽  
Anomaly Equation ◽  
Gravity Theories

We propose interpreting the zero curvature condition associated with an integrable model as an anomaly equation. This can lead to the WZWN action and the associated current algebra quite readily and clarifies further the connections found between the integrable models and 2D gravity theories. We analyze, in detail, the cases SL (2, R) (KdV hierarchy), OSp (2/1) (sKdV hierarchy) and SL (3, R) (Boussinesq hierarchy) and obtain the operator product expansions of the appropriate fields. We also make some observations on the generalization of our method to SL (n, R).

scholarly journals QUANTUM GROUP SYMMETRY OF INTEGRABLE SYSTEMS WITH OR WITHOUT BOUNDARY

2002 ◽  
Vol 17 (25) ◽  
pp. 3649-3661 ◽  
Author(s):  
E. RAGOUCY
Keyword(s):  
Quantum Group ◽  
Integrable Systems ◽  
Integrable Hierarchies ◽  
Group Symmetry ◽  
Integrals Of Motion ◽  
R Matrix ◽  
Explicit Expressions

We present a construction of integrable hierarchies without or with boundary, starting from a single R-matrix, or equivalently from a ZF algebra. We give explicit expressions for the Hamiltonians and the integrals of motion of the hierarchy in term of the ZF algebra. In the case without boundary, the integrals of motion form a quantum group, while in the case with boundary they form a Hopf coideal subalgebra of the quantum group.

scholarly journals Transport in perturbed classical integrable systems: The pinned Toda chain

2018 ◽  
Vol 117 ◽  
pp. 249-254 ◽  
Author(s):  
Pierfrancesco Di Cintio ◽  
Stefano Iubini ◽  
Stefano Lepri ◽  
Roberto Livi
Keyword(s):  
Integrable Systems ◽  
Toda Chain ◽  
Classical Integrable

scholarly journals LAX-PAIR FORMULATION OF THE W-GRAVITY THEORIES IN TWO DIMENSIONS

1992 ◽  
Vol 07 (27) ◽  
pp. 6907-6932 ◽  
Author(s):  
LINA JEAN-MARC ◽  
PRASANTA K. PANIGRAHI
Keyword(s):  
Light Cone ◽  
2D Gravity ◽  
Lax Pair ◽  
Two Dimensions ◽  
Two Dimensional ◽  
Curvature Condition ◽  
Induced Gravity ◽  
Light Cone Gauge ◽  
Zero Curvature ◽  
Cosmological Constants

The Lax-pair formulation of the two-dimensional induced gravity in the light-cone gauge is extended to the more general wN theories. After presenting the w2 and w3 gravities, we give a general prescription for an arbitrary wN case. This is further illustrated with the w4 gravity to point out some peculiarities. The constraints and the possible presence of the cosmological constants are systematically exhibited in the zero-curvature condition, which also yields the relevant Ward identities. The restrictions on the gauge parameters in the presence of the constraints are pointed out too, and are contrasted with those of the ordinary 2D gravity.

scholarly journals SL(2)k WZNW MODELS, PERTURBATIONS AND INTEGRABLE MODELS

1991 ◽  
Vol 06 (09) ◽  
pp. 1617-1639 ◽  
Author(s):  
L. BONORA ◽  
Y.-Z. ZHANG ◽  
M. MARTELLINI
Keyword(s):  
Conservation Laws ◽  
Linear Systems ◽  
Equations Of Motion ◽  
Integrable Models ◽  
Lax Pairs ◽  
R Matrix ◽  
Zero Curvature ◽  
Perturbed Systems ◽  
Wznw Model

Perturbations of the SL(2)kC WZNW model by relevant operators are studied. Nontrivial off-critical conservation laws in these perturbed systems are constructed. For the rational level k, one of the perturbations is shown to be an analogue of the (1, 2j+1) operator in the WZNW model. Linear systems (Lax pairs) are defined whose zero curvature conditions give rise to the classical versions of the perturbed equations of motion. The classical r-matrix for these linear systems is found.

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