2134/20232
Evgeny Ferapontov
Evgeny
Ferapontov
Jonathan Moss
Jonathan
Moss
Linearly degenerate PDEs and quadratic line complexes
Loughborough University
2016
Multi-dimensional second order PDE
Quadratic line complex
Linear degeneracy
Conformal structure
Integrability
Mathematical Sciences not elsewhere classified
2016-02-05 11:51:58
Journal contribution
https://repository.lboro.ac.uk/articles/journal_contribution/Linearly_degenerate_PDEs_and_quadratic_line_complexes/9385601
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.