scholarly journals Scaling invariance of the strict KP hierarchy

2020 ◽  
pp. 331-340
Author(s):  
Gerard F. Helminck ◽  
Elena A. Panasenko
Keyword(s):  
Standard Setting ◽  
Minimal Realization ◽  
Kp Hierarchy ◽  
Scaling Invariance ◽  
Lax Equations ◽  
Strict Kp Hierarchy

In this paper we show first of all that for solutions of the strict KP hierarchy it is sufficient to work in a standard setting. Further we discuss a minimal realization of the hierarchy and present the scaling invariance of the Lax equations of the hierarchy.

GENERALIZED N=2 SUPER KdV HIERACHIES: LIE SUPERALGEBRAIC METHODS AND SCALAR SUPER LAX FORMALISM

1992 ◽  
Vol 07 (supp01a) ◽  
pp. 419-447 ◽  
Author(s):  
TAKEO INAMI ◽  
HIROAKI KANNO
Keyword(s):  
Lie Superalgebras ◽  
Kp Hierarchy ◽  
Gauge Transformations ◽  
Lax Equations

Generalized N=2 super KdV hierarchies are constructed based on super Lax equations associated with Lie superalgebras SL(n|n)(1). The equivalence of the scalar super Lax formalism and the Lie superalgebraic method is derived by taking account of gauge transformations regarding the centre of SL(n|n)(1). We show that generalized N=2 super KdV hierarchies are related to the even reductions of the super KP hierarchy.

Reductions of the strict KP hierarchy

2020 ◽  
Vol 205 (2) ◽  
pp. 1411-1425
Author(s):  
G. F. Helminck ◽  
E. A. Panasenko
Keyword(s):  
Kp Hierarchy ◽  
Strict Kp Hierarchy

Geometric Solutions of the Strict KP Hierarchy

2019 ◽  
Vol 198 (1) ◽  
pp. 48-68 ◽  
Author(s):  
G. F. Helminck ◽  
E. A. Panasenko
Keyword(s):  
Kp Hierarchy ◽  
Geometric Solutions ◽  
Strict Kp Hierarchy

The Zero Curvature Form of Integrable Hierarchies in the ℤ × ℤ-Matrices

2012 ◽  
Vol 19 (02) ◽  
pp. 237-262
Author(s):  
G. F. Helminck ◽  
A. V. Opimakh
Keyword(s):  
Lie Algebra ◽  
Integrable Hierarchies ◽  
Evolution Equations ◽  
Integrable Hierarchy ◽  
Connection Form ◽  
Minimal Realization ◽  
Band Matrices ◽  
Zero Curvature ◽  
Lax Equations ◽  
Finite Band

In this paper it is shown how one can associate to a finite number of commuting directions in the Lie algebra of upper triangular ℤ × ℤ-matrices an integrable hierarchy consisting of a set of evolution equations for perturbations of the basic directions inside the mentioned Lie algebra. They amount to a tower of differential and difference equations for the coefficients of these perturbed matrices. The equations of the hierarchy are conveniently formulated in so-called Lax equations for these perturbations. They possess a minimal realization for which it is shown that the relevant evolutions of the perturbation commute. These Lax equations are shown in a purely algebraic way to be equivalent with zero curvature equations for a collection of finite band matrices, that are the components of a formal connection form. One concludes with the linearization of the hierarchies and the notion of wave matrices at zero, which is the algebraic substitute for a basis of the horizontal sections of the formal connection corresponding to this connection form.

Darboux transformations for the strict KP hierarchy

2021 ◽  
Vol 206 (3) ◽  
pp. 296-314
Author(s):  
G. F. Helminck ◽  
E. A. Panasenko
Keyword(s):  
Darboux Transformations ◽  
Kp Hierarchy ◽  
Strict Kp Hierarchy

Bilinear equations for the strict KP hierarchy

2015 ◽  
Vol 185 (3) ◽  
pp. 1803-1815 ◽  
Author(s):  
G. F. Helminck ◽  
E. A. Panasenko ◽  
S. V. Polenkova
Keyword(s):  
Kp Hierarchy ◽  
Bilinear Equations ◽  
Strict Kp Hierarchy
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