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Linearly degenerate PDEs and quadratic line complexes

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posted on 05.02.2016, 11:51 by Evgeny FerapontovEvgeny Ferapontov, Jonathan Moss
A quadratic line complex is a three-parameter family of lines in projective space P3 specified by a single quadratic relation in the Plücker coordinates. Fixing a point p in P3 and taking all lines of the complex passing through p we obtain a quadratic cone with vertex at p. This family of cones supplies P3 with a conformal structure, which can be represented in the form fij(p)dpidpj in a system of affine coordinates p = (p1; p2; p3). With this conformal structure we associate a three-dimensional second-order quasilinear wave equation, X i;j fij(ux1 ; ux2 ; ux3 )uxixj = 0; whose coefficients can be obtained from fij(p) by setting p1 = ux1 ; p2 = ux2 ; p3 = ux3 . We show that any PDE arising in this way is linearly degenerate, furthermore, any linearly degenerate PDE can be obtained by this construction. This provides a classi cation of linearly degenerate wave equations into eleven types, labelled by Segre symbols of the associated quadratic complexes. We classify Segre types for which the structure fij(p)dpidpj is conformally at, as well as Segre types for which the corresponding PDE is integrable.



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Communications in Analysis and Geometry


FERAPONTOV, E.V. and MOSS, J., 2015. Linearly degenerate PDEs and quadratic line complexes. Communications in Analysis and Geometry, 23(1), pp. 91–127.


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AM (Accepted Manuscript)

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This work is made available according to the conditions of the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International (CC BY-NC-ND 4.0) licence. Full details of this licence are available at: https://creativecommons.org/licenses/by-nc-nd/4.0/

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This paper was accepted for publication in the Communications in Analysis and Geometry. The definitive published version can be found at:http://dx.doi.org/10.4310/CAG.2015.v23.n1.a3