Supersymmetric KdV equation: Darboux transformation and discrete systems

We show that the supersymmetric nonlinear Schrödinger equation is a bi-Hamiltonian integrable system. We obtain the two Hamiltonian structures of the theory from the ones of the supersymmetric two-boson hierarchy through a field redefinition. We also show how the two Hamiltonian structures of the supersymmetric KdV equation can also be derived from a Hamiltonian reduction of the supersymmetric two-boson hierarchy.

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Nonholonomic deformation of coupled and supersymmetric KdV equations and Euler–Poincaré–Suslov method

Reviews in Mathematical Physics ◽

10.1142/s0129055x15500117 ◽

2015 ◽

Vol 27 (04) ◽

pp. 1550011 ◽

Cited By ~ 4

Author(s):

Partha Guha

Keyword(s):

Kdv Equation ◽

Infinite Dimensional ◽

Kdv Equations ◽

Finite Dimensional ◽

Euler Top ◽

Two Component ◽

Finite Dimensional Case ◽

Coupled Kdv Equations ◽

The Right ◽

Supersymmetric Kdv Equation

Recently, Kupershmidt [38] presented a Lie algebraic derivation of a new sixth-order wave equation, which was proposed by Karasu-Kalkani et al. [31]. In this paper, we demonstrate that Kupershmidt's method can be interpreted as an infinite-dimensional analogue of the Euler–Poincaré–Suslov (EPS) formulation. In a finite-dimensional case, we modify Kupershmidt's deformation of the Euler top equation to obtain the standard EPS construction on SO(3). We extend Kupershmidt's infinite-dimensional construction to construct a nonholonomic deformation of a wide class of coupled KdV equations, where all these equations follow from the Euler–Poincaré–Suslov flows of the right invariant L2 metric on the semidirect product group [Formula: see text], where Diff (S1) is the group of orientation preserving diffeomorphisms on a circle. We generalize our construction to the two-component Camassa–Holm equation. We also give a derivation of a nonholonomic deformation of the N = 1 supersymmetric KdV equation, dubbed as sKdV6 equation and this method can be interpreted as an infinite-dimensional supersymmetric analogue of the Euler–Poincaré–Suslov (EPS) method.

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Darboux transformation for the supersymmetric nonisospectral KdV equation

Modern Physics Letters B ◽

10.1142/s0217984921502559 ◽

2021 ◽

pp. 2150255

Author(s):

Li Chen ◽

Shu-Fang Deng

Keyword(s):

Darboux Transformation ◽

Kdv Equation ◽

Lax Pair ◽

The One

Darboux transformation for the supersymmetric nonisospectral KdV equation is investigated. Based on the Lax pair, we successively construct the one-fold, two-fold and three-fold Darboux tansformations for the supersymmetric nonisospectral KdV equation. Moreover, we present the [Formula: see text]-fold Darboux transformation in the form of superdeterminant.

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