Linearly degenerate partial differential equations and quadratic line complexes

A quadratic line complex is a three-parameter family of lines in projective space $\mathbb{P}^3$ specified by a single quadratic relation in the Plücker coordinates. Fixing a point ${\bf p}$ in $\mathbb{P}^3$ and taking all lines of the complex passing through ${\bf p}$ we obtain a quadratic cone with vertex at ${\bf p}$. This family of cones supplies $\mathbb{P}^3$ with a conformal structure, which can be represented in the form $f_{ij}({\bf p})\,dp^idp^j$ in a system of affine coordinates ${\bf p}=(p^1, p^2, p^3)$. With this conformal structure we associate a three-dimensional second-order quasilinear wave equation\[\sum _{i, j}f_{ij}(u_{x_1}, u_{x_2}, u_{x_3}) u_{x_ix_j}=0,\]whose coefficients can be obtained from $f_{ij}({\bf p})$ by setting $p^1=u_{x_1}, \ p^2=u_{x_2}, \ p^3=u_{x_3}$. We show that any partial differential equations (PDE) arising in this way is linearly degenerate, furthermore, any linearly degenerate PDE can be obtained by this construction. This provides a classification of linearly degenerate wave equations into 11 types, labelled by Segre symbols of the associated quadratic complexes. We classify Segre types for which the structure $f_{ij}({\bf p}) \, dp^idp^j$ is conformally flat, as well as Segre types for which the corresponding PDE is integrable.

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Published 24 November 2014