By F. A. E. Pirani, D. C. Robinson, W. F. Shadwick (auth.)
The target of this paper is to teach that the speculation of jet bundles offers the fitting environment for the examine of Backlund trans formations. those differences are used to resolve yes partial differential equations, rather non-linear evolution equations. in fact jets were hired for it slow within the concept of partial differential equations, yet up to now little use has been made up of them in purposes. in the mean time, great development has been made within the research of non-linear evolution equations. This paintings has been inspired by way of the dis covery of outstanding houses of a few such equations, for instance the life of soliton strategies and of limitless se quences of conservation legislation. one of the strategies devised to house those equations are the inverse scattering process and the Backlund transformation. In our opinion the jet package deal formula deals a unifying geometrical framework for only status the homes of non-linear evolution equations and the innovations used to house them, even supposing we don't think about all of those homes and strategies the following. The relevance of the idea of jet bundles lS that it legitimates the perform of in regards to the partial derivatives of box variables as self reliant amounts. considering Backlund trans formations require from the outset manipulation of those partial derivatives, and repeated shifts of perspective approximately which variables are depending on which, this geometrical environment clari fies and simplifies the thoughts concerned, and gives the chance of bringing coherence to various disparate results.
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Additional resources for Local Jet Bundle Formulation of Bäcklund Transformations: With Applications to Non-Linear Evolution Equations
We shall describe three different coordinate-free formulations of the integrability conditions, the first in terms of 34 exterior ideals of differential forms, the second ~n terms of modules of I-forms (Pfaffians), the last in terms of vector fields. Each of these formulations has advantages in particular applications. First we develop the necessary machinery, generalizing from section 2. First of all, the projection induces on J2(M,N 1) x N2 a naturally defined module prl*~2(M,Nl). Next, the projection may be extended to J2(M,N) 1 x N 2 -+ Jl(M,N ) 1 x N2 and then induces on J2(M,N 1) x N2 the module TI~* ~* ~1(M,N2).
Even when the integrability conditions for y's determined by ~ are not arbitrary. l-jet lies in the image of of f's. 7) for some choice In the interesting cases, it turns out not only that one can eliminate the y's from the integrability conditions, obtaining equations for the f's alone, but also that one can eliminate the z's from the integrability conditions and their prolongations, obtaining equations for the g's alone. 40 To formulate this in jet bundle language we need to generalize from section 2 the construction of a prolongation.
7). 7). One would therefore expect to be able to write these conditions in coordinate-free form as equations on J 2 (M,N 1 ) x N2 , where exactly the right combinations of coordinates for the representation of these functions and derivatives are available. We shall describe three different coordinate-free formulations of the integrability conditions, the first in terms of 34 exterior ideals of differential forms, the second ~n terms of modules of I-forms (Pfaffians), the last in terms of vector fields.