# Second order quasilinear PDEs and conformal structures in projective space

@article{Burovskiy2008SecondOQ, title={Second order quasilinear PDEs and conformal structures in projective space}, author={Pavel Burovskiy and Eugene V. Ferapontov and Sergey P. Tsarev}, journal={arXiv: Exactly Solvable and Integrable Systems}, year={2008} }

We investigate second order quasilinear equations of the form f_{ij} u_{x_ix_j}=0 where u is a function of n independent variables x_1, ..., x_n, and the coefficients f_{ij} are functions of the first order derivatives p^1=u_{x_1}, >..., p^n=u_{x_n} only. We demonstrate that the natural equivalence group of the problem is isomorphic to SL(n+1, R), which acts by projective transformations on the space P^n with coordinates p^1, ..., p^n. The coefficient matrix f_{ij} defines on P^n a conformal…

## 36 Citations

A Geometry for Second-Order PDEs and their Integrability, Part I

- Mathematics
- 2010

For the purpose of understanding second-order scalar PDEs and their hydrodynamic integrability, we introduce G-structures that are induced on hypersurfaces of the space of symmetric matrices…

Loughborough University Institutional Repository Linearly degenerate PDEs and quadratic line complexes

- Mathematics
- 2018

A quadratic line complex is a three-parameter family of lines in projective space P specified by a single quadratic relation in the Plücker coordinates. Fixing a point p in P and taking all lines of…

Integrable GL(2) Geometry and Hydrodynamic Partial Differential Equations

- Mathematics
- 2009

This article is a local analysis of integrable GL(2)-structures of degree 4. A GL(2)-structure of degree n corresponds to a distribution of rational normal cones over a manifold M of dimension (n+1).…

Dispersionless Hirota Equations and the Genus 3 Hyperelliptic Divisor

- MathematicsCommunications in Mathematical Physics
- 2019

Equations of dispersionless Hirota type $$\begin{aligned} F(u_{x_ix_j})=0 \end{aligned}$$ F ( u x i x j ) = 0 have been thoroughly investigated in mathematical physics and differential geometry. It…

Linearly degenerate PDEs and quadratic line complexes

- Mathematics
- 2015

A quadratic line complex is a three-parameter family of lines in projective space P3 specified by a single quadratic relation in the Plucker coordinates. Fixing a point p in P3 and taking all lines…

Generalized Lie-algebraic structures related to integrable dispersionless dynamical systems and their application

- MathematicsJournal of Mathematical Sciences and Modelling
- 2018

Our review is devoted to Lie-algebraic structures and integrability properties of an interesting class of nonlinear dynamical systems called the dispersionless heavenly equations, which were…

On integrability in Grassmann geometries: integrable systems associated with fourfolds in Gr(3,5)

- Mathematics
- 2015

Let Gr(d,n) be the Grassmannian of d ‐dimensional linear subspaces of an n ‐dimensional vector space Vn . A submanifold X⊂Gr(d,n) gives rise to a differential system Σ(X) that governs d ‐dimensional…

Nonlocal symmetries, conservation laws, and recursion operators of the Veronese web equation

- MathematicsJournal of Geometry and Physics
- 2019

Integrable Equations of the Dispersionless Hirota type and Hypersurfaces in the Lagrangian Grassmannian

- Mathematics
- 2007

We investigate integrable second-order equations of the formwhich typically arise as the Hirota-type relations for various (2 + 1)-dimensional dispersionless hierarchies. Familiar examples include…

Linearly degenerate partial differential equations and quadratic line complexes

- Mathematics
- 2015

A quadratic line complex is a three-parameter family of lines in projective space P specified by a single quadratic relation in the Plücker coordinates. Fixing a point p in P and taking all lines of…

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