scholarly journals Darboux Transformations for the KP Hierarchy in the Segal-Wilson Setting

2001 ◽  
Vol 53 (2) ◽  
pp. 278-309 ◽  
Author(s):  
G. F. Helminck ◽  
J. W. van de Leur
Keyword(s):  
Closed Form ◽  
Darboux Transformations ◽  
Kp Hierarchy ◽  
Geometric Data

AbstractIn this paper it is shown that inclusions inside the Segal-Wilson Grassmannian give rise to Darboux transformations between the solutions of the KP hierarchy corresponding to these planes. We present a closed form of the operators that procure the transformation and express them in the related geometric data. Further the associated transformation on the level of τ-functions is given.

Darboux transformations and Darboux coverings: some applications to the KP hierarchy

1998 ◽  
Vol 15 (4) ◽  
Author(s):  
Paolo Casati ◽  
GREGORIO Falqui ◽  
FRANCO Magri ◽  
Marco Pedroni ◽  
Jorge Zubelli
Keyword(s):  
Darboux Transformations ◽  
Kp Hierarchy

Closed-form representations of iterated Darboux transformations for the massless Dirac equation

2021 ◽  
Vol 36 (08n09) ◽  
pp. 2150064
Author(s):  
Axel Schulze-Halberg
Keyword(s):  
Dirac Equation ◽  
Closed Form ◽  
Scalar Potential ◽  
Higher Order ◽  
Two Dimensional ◽  
Darboux Transformations ◽  
First Order ◽  
Dirac Systems ◽  
Form Representation ◽  
Exactly Solvable

It is shown that first-order Darboux transformations for the two-dimensional massless Dirac equation with scalar potential and for the Schrödinger equation are the same up to a change of coordinates. As a consequence, we obtain a closed-form representation of iterated, higher-order Darboux transformations for our Dirac equation. We use the formalism to generate several new exactly-solvable Dirac systems through higher-order Darboux transformations.

Classical Darboux transformations and the KP hierarchy

2001 ◽  
Vol 17 (4) ◽  
pp. 1067-1074 ◽  
Author(s):  
F Lambert ◽  
I Loris ◽  
J Springael
Keyword(s):  
Darboux Transformations ◽  
Kp Hierarchy

Exact Eigensolutions for a Family of Nonuniform Rods With End Point Masses

2017 ◽  
Vol 139 (5) ◽  
Author(s):  
Lourdes Rubio ◽  
José Fernández-Sáez ◽  
Antonino Morassi
Keyword(s):  
Closed Form ◽  
Canonical Form ◽  
Longitudinal Vibration ◽  
Countable Family ◽  
Darboux Transformations ◽  
Closed Form Solutions ◽  
End Point ◽  
Sturm Liouville ◽  
Tip Mass

In this paper, new exact closed-form solutions for free longitudinal vibration of a one-parameter countable family of cantilever rods with one end tip mass are obtained. The analysis is based on the reduction of the equation governing the longitudinal vibration to the Sturm–Liouville canonical form and on the use of double Darboux transformations. The rods for which exact eigensolutions are provided are explicitly determined in terms of an initial rod with known closed-form eigensolutions. The method can be also extended to include longitudinally vibrating rods with tip mass at both ends.

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

DARBOUX TRANSFORMATIONS FOR N-DIMENSIONAL EFFECTIVE MASS SCHRÖDINGER EQUATIONS WITH HYPERSPHERICAL SYMMETRY

2007 ◽  
Vol 22 (19) ◽  
pp. 3293-3304 ◽  
Author(s):  
AXEL SCHULZE-HALBERG
Keyword(s):  
Schrödinger Equation ◽  
Closed Form ◽  
Effective Mass ◽  
Darboux Transformation ◽  
Spherical Symmetry ◽  
Schrodinger Equation ◽  
Arbitrary Order ◽  
Schrodinger Equations ◽  
Schrödinger Equations ◽  
Darboux Transformations

We construct Darboux transformations of arbitrary order for stationary Schrödinger equations with effective mass that exhibit spherical symmetry in N dimensions (hyperspherical symmetry). The Darboux transformation and the associated transformed Schrödinger equation are obtained in closed form.

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