» » Local Jet Bundle Formulation of Bäckland Transformations: With Applications to Non-Linear Evolution Equations (Mathematical Physics Studies) (Volume 1)

Local Jet Bundle Formulation of Bäckland Transformations: With Applications to Non-Linear Evolution Equations (Mathematical Physics Studies) (Volume 1) ePub download

by F.A.E. Pirani

  • Author: F.A.E. Pirani
  • ISBN: 9027710368
  • ISBN13: 978-9027710369
  • ePub: 1416 kb | FB2: 1624 kb
  • Language: English
  • Category: Physics
  • Publisher: Springer; Softcover reprint of the original 1st ed. 1979 edition (October 31, 1979)
  • Pages: 140
  • Rating: 4.1/5
  • Votes: 591
  • Format: txt lrf docx mobi
Local Jet Bundle Formulation of Bäckland Transformations: With Applications to Non-Linear Evolution Equations (Mathematical Physics Studies) (Volume 1) ePub download

These transformations are used to solve certain partial differential equations, particularly non-linear evolution . The aim of this paper is to show that the theory of jet bundles supplies the appropriate setting for the study of Backlund trans­ formations.

These transformations are used to solve certain partial differential equations, particularly non-linear evolution equations. Of course jets have been. These transformations are used to solve certain partial differential equations, particularly non-linear evolution equations. Of course jets have been employed for some time in the theory of partial differential equations, but so far little use has been made of them in applications. In the meanwhile, substantial progress has been made in the study of non-linear evolution equations.

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Interdisciplinary Mathematics Volumes I. R Hermann. Local Jet Bundle Formulation of Bäcklund Transformations. In our opinion the jet bundle formulation offers a unifying geometrical framework for under­ standing the properties of non-linear evolution equations and the techniques used to deal with them, although we do not consider all of these properties and techniques here. The relevance of the theory of jet bundles lS that it legitimates the practice of regarding the partial derivatives of field variables as independent quantities.

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As with any wave equation, these equations lead to two types of solution .

As with any wave equation, these equations lead to two types of solution: advanced potentials (which are related to the configuration of the sources at future points in time), and retarded potentials (which are related to the past configurations of the sources); the former are usually disregarded where the field is to analyzed from a causality perspective. Analogous to the tensor formulation, two objects, one for the field and one for the current, are introduced. is a linear transformation from the space of 2-forms to the space of (4 − 2)-forms defined by the metric in Minkowski space (in four dimensions even by any metric conformal to this metric).

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Applications to dierential equations: recursion operators . Passing to nonlocalities 6. Horizontal cohomology . 4This means that for any A-homomorphism f : P → Q one has γi(Q) ◦ Di(f ) Di−1(Diff+1 (f )) ◦ γi(P ). 10. Dk−1(P ) satisfying (∆(a))(b) −(∆(b))(a)

Applications to dierential equations: recursion operators . C-modules on dierential equations . Dk−1(P ) satisfying (∆(a))(b) −(∆(b))(a). We call ∆(a) the evaluation of the multiderivation ∆ at the element a ∈ A. Using this interpretation, dene by induction on k + l the operation ∧ : Dk(A) ⊗A Dl(P ) → Dk+l(P ) by setting. a ∧ p ap, a ∈ D0(A) A, p ∈ D0(P ) P, and.

P: analysis I-II, linear algebra, L: Dürr, Filipovic, Georgii, Merkl, Pruscha, Oppel, Winkler) Mathematical Methods (MM): Advanced partial differential equations: Fourier transform, distributions, Sobolev spaces, applications hyperbolic and parabolic equations, variational methods, applications to linear elliptic equations and non-linear equations in mathematical .

The aim of this paper is to show that the theory of jet bundles supplies the appropriate setting for the study of Backlund trans­ formations. These transformations are used to solve certain partial differential equations, particularly non-linear evolution equations. Of course jets have been employed for some time in the theory of partial differential equations, but so far little use has been made of them in applications. In the meanwhile, substantial progress has been made in the study of non-linear evolution equations. This work has been encouraged by the dis­ covery of remarkable properties of some such equations, for example the existence of soliton solutions and of infinite se­ quences of conservation laws. Among the techniques devised to deal with these equations are the inverse scattering method and the Backlund transformation. In our opinion the jet bundle formulation offers a unifying geometrical framework for under­ standing the properties of non-linear evolution equations and the techniques used to deal with them, although we do not consider all of these properties and techniques here. The relevance of the theory of jet bundles lS that it legitimates the practice of regarding the partial derivatives of field variables as independent quantities. Since Backlund trans­ formations require from the outset manipulation of these partial derivatives, and repeated shifts of point of view about which variables are dependent on which, this geometrical setting clari­ fies and simplifies the concepts involved, and offers the prospect of bringing coherence to a variety of disparate results.
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