## Another Look On Recursion Operators

Citations: | 8 - 5 self |

### BibTeX

@MISC{Marvan_anotherlook,

author = {Michal Marvan},

title = {Another Look On Recursion Operators},

year = {}

}

### OpenURL

### Abstract

. Recursion operators of partial differential equations are identified with Backlund auto-transformations of linearized diffieties. Relations to the classical concept and its recent Guthrie's generalization are discussed. Traditionally, a recursion operator of a PDE is a linear operator L acting on symmetries: if f is a symmetry then so is Lf . This provides a convenient way to generate infinite families of symmetries (Olver [13]). Standard reference is [14]. The presence of a recursion operator has been soon recognized as one of the attributes of the complete integrability. An exact and rich theory has been developed within the class of evolution equations (see Fokas [2] and references therein). Recursion operators are, as a rule, nonlocal and so may turn out to be the symmetries they generate (giving a powerful source of nonlocal symmetries [10, 5]). A remarkable problem of inverting a recursion operator motivated Guthrie [3] to a generalization, with consequences for generation of n...

### Citations

991 |
Applications of Lie Groups to Differential Equations
- Olver
- 1993
(Show Context)
Citation Context ...of a PDE is a linear operator L acting on symmetries: if f is a symmetry then so is Lf . This provides a convenient way to generate infinite families of symmetries (Olver [13]). Standard reference is =-=[14]-=-. The presence of a recursion operator has been soon recognized as one of the attributes of the complete integrability. An exact and rich theory has been developed within the class of evolution equati... |

52 |
Bäcklund Transformations and Their Applications
- Rogers, Shadwick
- 1982
(Show Context)
Citation Context ..., on the other side, dimM = 2, the one-dimensional horizontal cohomology group coincides with the group of conservation laws, and nontrivial abelian coverings may occur. Backlund transformations (see =-=[15]-=-) admit a very transparent description in the above terms: A Backlund transformation from a diffiety E 1 to a diffiety E 2 is simply a pair of coverings ~ E ! E 1 , ~ E ! E 2 with a common total space... |

48 |
1989] Nonlocal trends in the geom- etry of differential equations: Symmetries, conservation laws, and Backiund transformations
- Vinogradov
- 1978
(Show Context)
Citation Context ... equations (see Fokas [2] and references therein). Recursion operators are, as a rule, nonlocal and so may turn out to be the symmetries they generate (giving a powerful source of nonlocal symmetries =-=[10, 5]-=-). A remarkable problem of inverting a recursion operator motivated Guthrie [3] to a generalization, with consequences for generation of nonlocal symmetries [4]. His nonlocalities are no longer limite... |

46 |
The C-spectral sequence, Lagrangian formalism, and conservation laws. I. The linear theory. II. The non linear theory
- Vinogradov
- 1984
(Show Context)
Citation Context ...ear if equations (3) are linear. Linear isomorphism classes of linear one-dimensional coverings correspond to one-dimensional horizontal cohomology classes, cf. [5]. Let us recall the essentials from =-=[18]-=-. ANOTHER LOOK ON RECURSION OPERATORS 395 LetsE denote the noncommutative C 1 E-algebra of antisymmetric forms on E, with respect to the wedge product , let CsE denote the ideal of contact forms, i.e.... |

32 |
Evolution equations possessing infinitely many symmetries
- Olver
- 1977
(Show Context)
Citation Context ...onally, a recursion operator of a PDE is a linear operator L acting on symmetries: if f is a symmetry then so is Lf . This provides a convenient way to generate infinite families of symmetries (Olver =-=[13]-=-). Standard reference is [14]. The presence of a recursion operator has been soon recognized as one of the attributes of the complete integrability. An exact and rich theory has been developed within ... |

19 | On zero-curvature representations of partial differential equations
- Marvan
- 1992
(Show Context)
Citation Context ...l linear morphisms R ! V E over E. For methods to find abelian coverings over diffieties see [18, 5]. Hints of Section 4 may be helpful. Finding coverings of Section 5 is possible by the procedure of =-=[12]-=-. Step 2 consists of solving a system of equations in total derivatives, which is, in fact, a part of the defining system for symmetries of R. Therefore, virtually any software to compute generating f... |

17 |
Recursion operators and non-local symmetries
- Guthrie
- 1994
(Show Context)
Citation Context ...le, nonlocal and so may turn out to be the symmetries they generate (giving a powerful source of nonlocal symmetries [10, 5]). A remarkable problem of inverting a recursion operator motivated Guthrie =-=[3]-=- to a generalization, with consequences for generation of nonlocal symmetries [4]. His nonlocalities are no longer limited to mere inverses D \Gamma1 of total derivatives. It seems to be an interestin... |

14 |
1992] Some new cohomological invariants for nonlinear differential equations
- Krasil'shchik
(Show Context)
Citation Context ...of total derivatives. It seems to be an interesting problem to give an interpretation of recursion operators in terms of the Vinogradov [17] category of diffieties. Recently, Krasilshchik and Kersten =-=[6, 8]-=- established a fundamental relation between recursion operators and deformations of diffiety structures. In this paper (Section 3) we propose another answer, according to which recursion operators are... |

11 |
Nonlocal symmetries of the KdV
- Guthrie, Hickman
- 1993
(Show Context)
Citation Context ...erful source of nonlocal symmetries [10, 5]). A remarkable problem of inverting a recursion operator motivated Guthrie [3] to a generalization, with consequences for generation of nonlocal symmetries =-=[4]-=-. His nonlocalities are no longer limited to mere inverses D \Gamma1 of total derivatives. It seems to be an interesting problem to give an interpretation of recursion operators in terms of the Vinogr... |

10 |
Symmetries of partial differential equations
- Vinogradov, ed
- 1989
(Show Context)
Citation Context ... nonlocalities are no longer limited to mere inverses D \Gamma1 of total derivatives. It seems to be an interesting problem to give an interpretation of recursion operators in terms of the Vinogradov =-=[17]-=- category of diffieties. Recently, Krasilshchik and Kersten [6, 8] established a fundamental relation between recursion operators and deformations of diffiety structures. In this paper (Section 3) we ... |

7 |
On the equivalence of linearization and formal symmetries as integrability tests for evolution equations
- Bilge
- 1993
(Show Context)
Citation Context ...on p + ; p \Gamma , respectively. The coverings P + ; P \Gamma are non-isomorphic. 7. Finding recursion operators Procedures to find recursion operators have been proposed by many authors, see, e.g., =-=[1, 8, 14]-=-. See also, e.g., [2] for deriving recursion operators from Lax pairs. Finding finitary linear recursion operators on the basis of the mere definition is, however, also possible. Steps to be followed ... |

6 |
Homological method of computing invariants of systems of differential equations
- Tsujishita
- 1991
(Show Context)
Citation Context ...dle of vertical tangent vectors to a diffiety E over M and by �� E the projection V E ! E. Then V E is a diffiety in a natural way, �� E is a covering, and V is a functor DEM ! DEM (see [11] a=-=nd also [10, 7, 16]). W-=-e give a self-contained exposition below. We start with two canonical ways of lifting functions from E to V E. The first is f 7! f ffi �� E , the second is f 7!sf , wheresf 2 C 1 V E satisfiessf f... |

5 |
Conservation laws and nonlocal symmetries
- Khorkova
(Show Context)
Citation Context ... equations (see Fokas [2] and references therein). Recursion operators are, as a rule, nonlocal and so may turn out to be the symmetries they generate (giving a powerful source of nonlocal symmetries =-=[10, 5]-=-). A remarkable problem of inverting a recursion operator motivated Guthrie [3] to a generalization, with consequences for generation of nonlocal symmetries [4]. His nonlocalities are no longer limite... |

3 |
A note on the category of partial differential equations
- Marvan
- 1987
(Show Context)
Citation Context ...ficient. (2) Obviously, (9) determines a zero-curvature representation of the equation E (is a Maurer--Cartan condition). Moreover, upon introducing extended matrices A (i) := 0 @ A (i) B (i) 0 0 1 A =-=(11)-=- (one column (B k (i) ) from the right and one row of zeroes from below) conditions (9, 10) reduce to a single Maurer--Cartan condition D j A (i) \Gamma D i A (j) + A (i) A (j) \Gamma A (j) A (i) = 0,... |

1 |
Deformations of differential equations and recursion operators
- Krasilshchik, Kersten
- 1993
(Show Context)
Citation Context ...of total derivatives. It seems to be an interesting problem to give an interpretation of recursion operators in terms of the Vinogradov [17] category of diffieties. Recently, Krasilshchik and Kersten =-=[6, 8]-=- established a fundamental relation between recursion operators and deformations of diffiety structures. In this paper (Section 3) we propose another answer, according to which recursion operators are... |