## Deformation and Recursion for the N = 2 α = 1 Supersymmetric KdV-hierarchy (2002)

### BibTeX

@MISC{Sorin02deformationand,

author = {Er S. Sorin and Paul H. M. Kersten},

title = {Deformation and Recursion for the N = 2 α = 1 Supersymmetric KdV-hierarchy},

year = {2002}

}

### OpenURL

### Abstract

Abstract. A detailed description is given for the construction of the deformation of the N = 2 supersymmetric α = 1 KdV-equation, leading to the recursion operator for symmetries and the zero-th Hamiltonian structure; the solution to a longstanding problem. 1.

### Citations

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Citation Context ...uux + uxxx. (1) For a short theoretical introduction we refer to [12], while for more detailed expositions we refer to [10, 7, 11]. We consider Y ⊂ J ∞ (x, t; u) the infinite prolongation of (1), c.f.=-=[13, 14]-=-, where coordinates in the infinite jet bundle J ∞ (x, t; u) are given by (x, t, u, ux, ut, · · · ) and Y is formally described as the submanifold of J ∞ (x, t; u) defined by ut = uux + uxxx, uxt = uu... |

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Citation Context ...s their nonpolynomiality. A new approach to a recursion operator treating it as a form–valued vector field which satisfies a generalized symmetry equation related to a given equation was developed in =-=[10, 11]-=-. Using this approach the recursion operator of the bosonic limit of the N=2 α = 1 KdV-hierarchy was derived in [12], and its structure, underlining relevance of these Hamiltonians in the bosonic limi... |

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Citation Context ...cursion operator for symmetries and the zero-th Hamiltonian structure; the solution to a longstanding problem. 1. Introduction The N = 2 supersymmetric α = 1 KdV-equation was originally introduced in =-=[1]-=- as a Hamiltonian equation with the N = 2 superconformal algebra as a second Hamiltonian structure, and its integrability was conjectured there due to the existence of a few additional nontrivial boso... |

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Citation Context ... {F = 0} is defined as a vertical vector field V where f ∈ C ∞ (Y ) are such that V = Зf = f∂u + ˜ Dx(f)∂u1 + ˜ D 2 x(f)∂u2 + . . . (4) ℓF(f) = 0. (5) Here, ℓF is the universal linearisation operator =-=[15, 13]-=- which is Fréchet derivative of F, and it reads in the case of the KdV-equation (2) ˜Dt(f) − u ˜ Dx(f) − u1 · f − ( ˜ Dx) 3 (f) = 0. (6) Let now W ⊂ Rm with coordinates (w1, · · · wm). The Cartan dist... |

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Citation Context ...Hamiltonian equations with the N = 2 superconformal algebra as a second Hamiltonian structure (the N = 2 α = −2 and α = 4 KdV-equations [3, 1]), but the N = 2 α = 1 KdV-equation is rather exceptional =-=[4]-=-. Despite knowledge of its Lax–pair description, there remains a lot of longstanding, unsolved problems which resolution would be quite important for a deeper understanding and more detailed descripti... |

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Citation Context ...es a generalized symmetry equation related to a given equation was developed in [10, 11]. Using this approach the recursion operator of the bosonic limit of the N=2 α = 1 KdV-hierarchy was derived in =-=[12]-=-, and its structure, underlining relevance of these Hamiltonians in the bosonic limit, gives a hint towards its supersymmetric generalization. The organisation of this paper is as follows. First the g... |

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Citation Context ...quation there are another two inequivalent N = 2 supersymmetric Hamiltonian equations with the N = 2 superconformal algebra as a second Hamiltonian structure (the N = 2 α = −2 and α = 4 KdV-equations =-=[3, 1]-=-), but the N = 2 α = 1 KdV-equation is rather exceptional [4]. Despite knowledge of its Lax–pair description, there remains a lot of longstanding, unsolved problems which resolution would be quite imp... |

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Citation Context ...onian structure, and its integrability was conjectured there due to the existence of a few additional nontrivial bosonic Hamiltonians. Then its Lax– pair representation has indeed been constructed in =-=[2]-=-, and it allowed an algoritmic reconstruction of the whole tower of highest commutative bosonic flows and their Hamiltonians belonging to the N = 2 supersymmetric α = 1 KdV-hierarchy. Actually, beside... |

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Citation Context ...phrases. Complete Integrability, Deformations, Bi-Hamiltonian Structure, Recursion Operators, Symmetries, Conservation Laws, Coverings. 12 ALEXANDER S. SORIN AND PAUL H. M. KERSTEN were uncovered in =-=[9]-=-. A new property, crucial for the existence of these flows and Hamiltonians, making them distinguished compared to all flows and Hamiltonians of other supersymmetric hierarchies constructed before, is... |

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Citation Context ...made to construct a tower of its noncommutative bosonic and fermionic, local and nonlocal symmetries and Hamiltonians, bi-Hamiltonian structure as well as recursion operator (see, e.g. discussions in =-=[5, 6]-=- and references therein). Though these rather complicated problems, solved for the case of the N = 2 α = −2 and α = 4 KdV-hierarchies, still wait their complete resolution for the N = 2 α = 1 KdV- hie... |

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Citation Context ...s their nonpolynomiality. A new approach to a recursion operator treating it as a form–valued vector field which satisfies a generalized symmetry equation related to a given equation was developed in =-=[10, 11]-=-. Using this approach the recursion operator of the bosonic limit of the N=2 α = 1 KdV-hierarchy was derived in [12], and its structure, underlining relevance of these Hamiltonians in the bosonic limi... |

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Edited and with a preface by Krasil ′ shchik and Vinogradov
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Citation Context ...ons governing the nonlocal variables wj (j = 1, . . . , m); i.e., (wj)x = X j , (wj)t = T j . The construction of the associated ∂wj (j = 1, . . . , m) components is called the reconstruction problem =-=[16]-=-. For reasons of simplicity, we omit this reconstruction problem, i.e. reconstructing the complete vector field or full symmetry from its shadow. The classical Lenard recursion operator R for the KdV-... |

4 | Symmetries and recursions for N = 2 supersymmetric KdV-equation, in Integrable Hierarchies and Modern Physical Theories, Eds - Kersten |

4 | Superalgebraic structure of the N=2 supersymmetric α = 1 KdV hierarchy, in preparation. Bogoliubov Laboratory of Theoretical - Kersten, Sorin |

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Open problems for the super KdV equation. AARMS-CRM Workshop on Baecklund and Darboux transformations. The Geometry of Soliton Theory. June 4–9
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Citation Context ...made to construct a tower of its noncommutative bosonic and fermionic, local and nonlocal symmetries and Hamiltonians, bi-Hamiltonian structure as well as recursion operator (see, e.g. discussions in =-=[5, 6]-=- and references therein). Though these rather complicated problems, solved for the case of the N = 2 α = −2 and α = 4 KdV-hierarchies, still wait their complete resolution for the N = 2 α = 1 KdV- hie... |