By R. J. Alonso, S. Jimènez, J. Rodríguez (auth.), Boris Kruglikov, Valentin Lychagin, Eldar Straume (eds.)

ISBN-10: 3642008720

ISBN-13: 9783642008726

The Abel Symposium 2008 serious about the fashionable conception of differential equations and their functions in geometry, mechanics, and mathematical physics. Following the culture of Monge, Abel and Lie, the clinical application emphasised the function of algebro-geometric tools, which these days permeate all mathematical types in typical and engineering sciences. the information of invariance and symmetry are of basic value within the geometric method of differential equations, with a significant influence coming from the world of integrable platforms and box theories.

This quantity involves unique contributions and vast review lectures of the contributors of the Symposium. The papers during this quantity current the fashionable method of this classical subject.

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Additional info for Differential Equations - Geometry, Symmetries and Integrability: The Abel Symposium 2008

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We may view measures as n-forms of odd type [dRh55], hence freely consider the pullback measure φ ∗ ν of a measure ν by a map φ : M → M. Pulling-back does not preserve the total mass: M d(φ ∗ ν) ≡ M dν, unless φ is a diffeomorphism. This is in contrast with the push-forward (also called transport) of a measure μ by a map φ : M → M, denoted by φ# μ, which may be defined (via the Riesz representation theorem [Rie09]) by: u dν := M (u ◦ φ) dμ, with ν = φ# μ, (1) M where u stands for an arbitrary continuous real function on M.

8) The orbits of G are two dimensional so that the quotient space M/G has dimension two with coordinates (say) {u, v }. The invariants for action of G on M are y1 and y3 /y22 so that we may write the projection q as u = y1 , v= y3 . y22 (9) By solving the linear system X (a1 θ 1 + a2 θ 2 + a3 θ 3 + a4 θ 4 ) = 0 for all X ∈ (10) we determine that Isb = { θ 4 − y2 y2 2y3 3 2y3 θ + 32 θ 2 } = { dy3 − dy2 + 32 dy1 }. y2 y2 3y2 3y2 (11) We substitute y3 = vy22 and y1 = u into this result to find that the reduced EDS is I/G = { v 2 du + 3dv } and the reduced ODE is (12) 1 dv = − v2.

L. Bryant, P. A. Griffiths, and L. , New series 1 (1995), 21–122 and 265–323. ´ Cartan, Sur int´egration des syst`emes diff´erentiels compl`etement int´egrables, C. R. Acad. 8. E. Sc, 134 (1902), 1415–1418 and 1564–1566. 9. E´ Cartan, Les syst`emes de Pfaff a` cinq variables et les e´ quations aux d´eriv´ees partielles du ´ second ordre, Ann. Sci. Ecole Norm. 3 (1910), no. 27, 109–192. 10. J. N. Clelland and T. A. Ivey, B¨acklund transformatiosn and Darboux integrability for nonlinear wave equations, to appear, Asian J.

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Differential Equations - Geometry, Symmetries and Integrability: The Abel Symposium 2008 by R. J. Alonso, S. Jimènez, J. Rodríguez (auth.), Boris Kruglikov, Valentin Lychagin, Eldar Straume (eds.)


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