Generalized intermediate long‐wave hierarchy in zero‐curvature representation with noncommutative spectral parameter

1992 ◽  
Vol 33 (11) ◽  
pp. 3783-3793 ◽  
Author(s):  
A. Degasperis ◽  
D. Lebedev ◽  
M. Olshanetsky ◽  
S. Pakuliak ◽  
A. Perelomov ◽  
...  
Keyword(s):  
Spectral Parameter ◽  
Zero Curvature Representation ◽  
Zero Curvature ◽  
Long Wave

Classically Integrable Non-Linear Sigma Models and their Geometric Properties

2021 ◽  
Vol 59 ◽  
pp. 47-65
Author(s):  
Paul Bracken
Keyword(s):  
Sigma Models ◽  
Spectral Parameter ◽  
Equations Of Motion ◽  
Mathematical Structure ◽  
Geometrical Properties ◽  
Zero Curvature Representation ◽  
Zero Curvature ◽  
New Models ◽  
Non Linear ◽  
Linear Sigma Models

General classes of non-linear sigma models originating from a specified action are developed and studied. Models can be grouped and considered within a single mathematical structure this way. The geometrical properties of these models and the theories they describe are developed in detail. The zero curvature representation of the equations of motion are found. Those representations which have a spectral parameter are of importance here. Some new models with Lax pairs which depend on a spectral parameter are found. Some particular classes of solutions are worked out and discussed.

scholarly journals A Family of Integrable Differential-Difference Equations: Tri-Hamiltonian Structure and Lie Algebra of Vector Fields

2021 ◽  
Vol 2021 ◽  
pp. 1-9
Author(s):  
Ning Zhang ◽  
Xi-Xiang Xu
Keyword(s):  
Spectral Problem ◽  
Lie Algebra ◽  
Difference Equations ◽  
Vector Fields ◽  
Hamiltonian Structure ◽  
Trace Identity ◽  
Zero Curvature Representation ◽  
Curvature Equation ◽  
Zero Curvature ◽  
Liouville Integrable

Starting from a novel discrete spectral problem, a family of integrable differential-difference equations is derived through discrete zero curvature equation. And then, tri-Hamiltonian structure of the whole family is established by the discrete trace identity. It is shown that the obtained family is Liouville-integrable. Next, a nonisospectral integrable family associated with the discrete spectral problem is constructed through nonisospectral discrete zero curvature representation. Finally, Lie algebra of isospectral and nonisospectral vector fields is deduced.

ON SIGMA MODEL FORMULATION OF GREEN-SCHWARZ SUPERSTRING

1989 ◽  
Vol 04 (04) ◽  
pp. 351-359 ◽  
Author(s):  
A. ISAEV ◽  
E. IVANOV
Keyword(s):  
Direct Product ◽  
Sigma Model ◽  
Model Formulation ◽  
Field Equations ◽  
Zero Curvature Representation ◽  
Zero Curvature ◽  
Specific Choice ◽  
Target Manifold

The Green-Schwarz covariant superstring action is consistently deduced as the action of the Wess-Zumino-Witten σ-model defined on the direct product of two N = 1, D = 10 Poincaré supertranslation groups. N = 2 supersymmetry of the action is shown to be related to a specific choice of the target manifold. We propose a zero curvature representation for the GS superstring field equations and interpret the local fermionic supersymmetry of the GS action as a guage symmetry preserving this representation.

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