## Available via INTERNET: (2001)

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author = {},

title = {Available via INTERNET:},

year = {2001}

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### Abstract

On the horizontal gauge cohomology and non-removability of the spectral parameter by

### Citations

252 |
Nonlinear Evolution Equations and Inverse Scattering., volume 244. Cambridge Univ
- Ablowitz, Segur
- 1992
(Show Context)
Citation Context ...uge cohomology group ¯ H 1 α is shown to contain obstructions to removability of the “spectral parameter” of α. is suggested. compute ¯H 1 α A method to Introduction Important integration techniques (=-=[1, 6, 8, 9]-=-) for soliton equations, especially in dimension two, involve a zero-curvature representation (ZCR) taking valuesin a non-solvable Lie algebra g and depending on a non-removable “spectral” parameter (... |

63 |
Integrable Models, World Scientific
- Das
- 1989
(Show Context)
Citation Context ...uge cohomology group ¯ H 1 α is shown to contain obstructions to removability of the “spectral parameter” of α. is suggested. compute ¯H 1 α A method to Introduction Important integration techniques (=-=[1, 6, 8, 9]-=-) for soliton equations, especially in dimension two, involve a zero-curvature representation (ZCR) taking valuesin a non-solvable Lie algebra g and depending on a non-removable “spectral” parameter (... |

59 |
Prolongation Structures of Nonlinear Evolution
- Estabrook, Wahlquist
- 1976
(Show Context)
Citation Context ...ete integrability” of nonlinear partial differential equations(PDE). From another perspective, ZCR’s are linear, finite-dimensional coverings[15], and often realizationsof the Wahlquist and Estabrook =-=[25]-=- prolongation structures. Yet another interpretation has been given in terms of pseudospherical surfaces [19, 10]. Relations to the geometry of surfaces are also the subject of [5, 20, 21]. In [16], u... |

48 |
1989] Nonlocal trends in the geom- etry of differential equations: Symmetries, conservation laws, and Backiund transformations
- Vinogradov
- 1978
(Show Context)
Citation Context ...ear problems”, are often considered as attributes of “complete integrability” of nonlinear partial differential equations(PDE). From another perspective, ZCR’s are linear, finite-dimensional coverings=-=[15]-=-, and often realizationsof the Wahlquist and Estabrook [25] prolongation structures. Yet another interpretation has been given in terms of pseudospherical surfaces [19, 10]. Relations to the geometry ... |

46 |
The C-spectral sequence, Lagrangian formalism and conservation
- Vinogradov
- 1984
(Show Context)
Citation Context ...h coefficientsin a larger algebra. It should be noted that when the Lie algebra g isone-dimensional, then the horizontal gauge cohomology becomesthe horizontal cohomology ¯ Hq = E 0,q 1 of Vinogradov =-=[23]-=- and iseffectively computable by meansof the C-spectral sequence [23, 22]. Also, according to Verbovetsky [24], for general g we still have ¯ Hq α = E 0,q 1 for a generalized C-spectral sequence with ... |

27 |
Groupes et algèbres de Lie, Éléments de mathématique
- Bourbaki
(Show Context)
Citation Context ...σ, ρ], ¯ d[ρ, σ]=[ ¯ dρ, σ]+(−1) r [ρ, ¯ dσ] for ρ ∈ ¯ Λ r E⊗g, σ ∈ ¯ Λ s E⊗g. It is technically convenient to assume that g is a matrix algebra (the assumption being irrestrictive by the Ado theorem =-=[3]-=-), i.e., that g is a subalgebra in some gl n.Then¯ΛE⊗gl n isa graded associative algebra with respect to the multiplication Aµ·Bν = (A · B)µ ∧ ν induced by the ordinary matrix multiplication, while [ρ... |

19 | On zero-curvature representations of partial differential equations
- Marvan
- 1992
(Show Context)
Citation Context ...ok [25] prolongation structures. Yet another interpretation has been given in terms of pseudospherical surfaces [19, 10]. Relations to the geometry of surfaces are also the subject of [5, 20, 21]. In =-=[16]-=-, using the so-called first gauge cohomology, a characteristic element χα was associated to any ZCR α of a formally integrable system of nonlinear PDE’s; the case of g = sl2 being further developed in... |

15 |
On zero-curvature representations of evolution equations
- Sakovich
- 1995
(Show Context)
Citation Context ...lly integrable system of nonlinear PDE’s; the case of g = sl2 being further developed in [17]. Independently, in his treatment of “continual classes” of evolution equations possessing a ZCR, Sakovich =-=[18]-=- introduced essentially the same concept, although without cohomological interpretation. In thispaper we deal with zeroth (horizontal) gauge cohomology ¯ H q α associated with any ZCR α (Section 1). F... |

14 |
1992] Some new cohomological invariants for nonlinear differential equations
- Krasil'shchik
(Show Context)
Citation Context ...paper is to a great extent parallel to the work by Krasil’shchik and Igonin [13], who study general coverings depending on a parameter, with the socalled C-cohomology in the background (Krasil’shchik =-=[11, 12]-=-). 2000 Mathematics Subject Classification. 35A30, 58G05. Key words and phrases. Zero-curvature representation, horizontal cohomology, spectral parameter, pseudospherical surface. 12 MICHAL MARVAN ZC... |

13 |
Nonlinear evolution equations of physical significance
- Ablowitz, Kaup, et al.
(Show Context)
Citation Context ...) for soliton equations, especially in dimension two, involve a zero-curvature representation (ZCR) taking valuesin a non-solvable Lie algebra g and depending on a non-removable “spectral” parameter (=-=[2, 26]-=-). This is why oneparametric families of ZCR’s, also referred to as “linear problems”, are often considered as attributes of “complete integrability” of nonlinear partial differential equations(PDE). ... |

13 |
Pseudospherical surfaces and evolution equations
- Chern, Tenenblat
- 1986
(Show Context)
Citation Context ...uist and Estabrook [25] prolongation structures. Yet another interpretation has been given in terms of pseudospherical surfaces [19, 10]. Relations to the geometry of surfaces are also the subject of =-=[5, 20, 21]-=-. In [16], using the so-called first gauge cohomology, a characteristic element χα was associated to any ZCR α of a formally integrable system of nonlinear PDE’s; the case of g = sl2 being further dev... |

13 |
Transformation of manifolds and applications to differential equations
- Tenenblat
- 1998
(Show Context)
Citation Context ...uist and Estabrook [25] prolongation structures. Yet another interpretation has been given in terms of pseudospherical surfaces [19, 10]. Relations to the geometry of surfaces are also the subject of =-=[5, 20, 21]-=-. In [16], using the so-called first gauge cohomology, a characteristic element χα was associated to any ZCR α of a formally integrable system of nonlinear PDE’s; the case of g = sl2 being further dev... |

12 |
A.: The prolongation structures of quasipolynomial flows
- Dodd, Fordy
- 1983
(Show Context)
Citation Context ...uge cohomology group ¯ H 1 α is shown to contain obstructions to removability of the “spectral parameter” of α. is suggested. compute ¯H 1 α A method to Introduction Important integration techniques (=-=[1, 6, 8, 9]-=-) for soliton equations, especially in dimension two, involve a zero-curvature representation (ZCR) taking valuesin a non-solvable Lie algebra g and depending on a non-removable “spectral” parameter (... |

10 |
On differential equations describing pseudo -spherical surfaces
- Kamran, Tenenblat
- 1995
(Show Context)
Citation Context ...ar, finite-dimensional coverings[15], and often realizationsof the Wahlquist and Estabrook [25] prolongation structures. Yet another interpretation has been given in terms of pseudospherical surfaces =-=[19, 10]-=-. Relations to the geometry of surfaces are also the subject of [5, 20, 21]. In [16], using the so-called first gauge cohomology, a characteristic element χα was associated to any ZCR α of a formally ... |

9 |
Soliton Surfaces and their Applications
- Sym
- 1985
(Show Context)
Citation Context ...uist and Estabrook [25] prolongation structures. Yet another interpretation has been given in terms of pseudospherical surfaces [19, 10]. Relations to the geometry of surfaces are also the subject of =-=[5, 20, 21]-=-. In [16], using the so-called first gauge cohomology, a characteristic element χα was associated to any ZCR α of a formally integrable system of nonlinear PDE’s; the case of g = sl2 being further dev... |

7 |
Conservation laws for nonlinear evolution equations
- Cavalcante, Tenenblat
- 1988
(Show Context)
Citation Context ...s, see the references in [21]. Needless to say, no infinite series of conservation laws will be generated in the case of removable parameter. Here and in the appendix below we complete the results of =-=[4, 5]-=- by showing that exactly this happens in the example of the Burgersequation. Writing the Burgersequation asut = uxx + uux, authorsof [4] consider the ZCR (9) αη = Aη dx + Bη dt = + ( 1 4ux + 1 ( 1 2η ... |

7 |
On one-parametric families of Bäcklund transformations. Lie Groups, Geometric Structures and Differential Equations—One Hundred Years After Sophus Lie
- Krasil’shchik, Igonin
(Show Context)
Citation Context ...duce a horizontal cohomology class [ ˙α] ∈ ¯H 1 α that is an obstruction to removability of the parameter (Section 2). This paper is to a great extent parallel to the work by Krasil’shchik and Igonin =-=[13]-=-, who study general coverings depending on a parameter, with the socalled C-cohomology in the background (Krasil’shchik [11, 12]). 2000 Mathematics Subject Classification. 35A30, 58G05. Key words and ... |

7 |
Soliton equations and pseudospherical surfaces
- Sasaki
- 1979
(Show Context)
Citation Context ...ar, finite-dimensional coverings[15], and often realizationsof the Wahlquist and Estabrook [25] prolongation structures. Yet another interpretation has been given in terms of pseudospherical surfaces =-=[19, 10]-=-. Relations to the geometry of surfaces are also the subject of [5, 20, 21]. In [16], using the so-called first gauge cohomology, a characteristic element χα was associated to any ZCR α of a formally ... |

6 |
Homological method of computing invariants of systems of differential equations
- Tsujishita
- 1991
(Show Context)
Citation Context ...ie algebra g isone-dimensional, then the horizontal gauge cohomology becomesthe horizontal cohomology ¯ Hq = E 0,q 1 of Vinogradov [23] and iseffectively computable by meansof the C-spectral sequence =-=[23, 22]-=-. Also, according to Verbovetsky [24], for general g we still have ¯ Hq α = E 0,q 1 for a generalized C-spectral sequence with coefficients in a C-module, which leadsto a computation method for the te... |

5 |
Integrirovanie nelinejnykh uravnenij matematicheskoj fiziki metodom obratnoj zadachi rasseyaniya
- Zakharov, Shabat
- 1979
(Show Context)
Citation Context ...) for soliton equations, especially in dimension two, involve a zero-curvature representation (ZCR) taking valuesin a non-solvable Lie algebra g and depending on a non-removable “spectral” parameter (=-=[2, 26]-=-). This is why oneparametric families of ZCR’s, also referred to as “linear problems”, are often considered as attributes of “complete integrability” of nonlinear partial differential equations(PDE). ... |

4 | Notes on the horizontal cohomology
- Verbovetsky
- 1997
(Show Context)
Citation Context ...horizontal gauge cohomology becomesthe horizontal cohomology ¯ Hq = E 0,q 1 of Vinogradov [23] and iseffectively computable by meansof the C-spectral sequence [23, 22]. Also, according to Verbovetsky =-=[24]-=-, for general g we still have ¯ Hq α = E 0,q 1 for a generalized C-spectral sequence with coefficients in a C-module, which leadsto a computation method for the terms E p,q 1 with p ≥ 1, but not the t... |

2 |
A direct method to compute zero curvature representations. The case sl2
- Marvan
(Show Context)
Citation Context ...nt nonlocal conservation laws. Likewise, α degenerates(isgauge equivalent to the one that satisfies a12b21 = a21b12) whenever at least one of the off-diagonal coefficients a12,a21,b12,b21 iszero (see =-=[17]-=-). Let C = χ1 or any other χi that isnonzero (if all are zero, then α is trivial). Assuming α non-degenerate, by the main result of [17] we have exactly two possible normal forms for the matrices C, A... |

1 |
Linear Systems of Ordinary Differential Equations
- Erugin
- 1966
(Show Context)
Citation Context ...e coboundaries, find g-matrices Qλ such that ˙αλ = ¯∂αλ Qλ. (3) Solve the equation ∂Sλ/∂λ = QλSλ for a G-matrix Sλ under the initial condition Sλ = E. For methods to perform the third step see, e.g., =-=[7]-=-. Concerning the second step, we shall show in the next section that H 1 α iseffectively computable in principle. But before that we present an example. Example 4. Sasaki [19] was probably the first t... |

1 |
Cohomology background in geometry of PDE
- Krasil’shchik
- 1998
(Show Context)
Citation Context ...paper is to a great extent parallel to the work by Krasil’shchik and Igonin [13], who study general coverings depending on a parameter, with the socalled C-cohomology in the background (Krasil’shchik =-=[11, 12]-=-). 2000 Mathematics Subject Classification. 35A30, 58G05. Key words and phrases. Zero-curvature representation, horizontal cohomology, spectral parameter, pseudospherical surface. 12 MICHAL MARVAN ZC... |