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… 
A Geometry for Second-Order PDEs and their Integrability, Part I
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
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
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
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
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
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)
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
Integrable Equations of the Dispersionless Hirota type and Hypersurfaces in the Lagrangian Grassmannian
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
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
...
...

References

SHOWING 1-10 OF 82 REFERENCES
Integrable Lagrangians and modular forms
Dispersionless Limit of Integrable Systems in 2 + 1 Dimensions
A general scheme for construction of dispersionless limits of 2 + 1 dimensional integrable systems was described first in the article [1]. Now we give its description in more details. Let us consider
On a Class of Three-Dimensional Integrable Lagrangians
AbstractWe characterize non-degenerate Lagrangians of the form such that the corresponding Euler-Lagrange equations are integrable by the method of hydrodynamic reductions. The integrability
Integrable Equations of the Dispersionless Hirota type and Hypersurfaces in the Lagrangian Grassmannian
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
Moduli spaces of quadratic complexes and their singular surfaces
We construct the coarse moduli space $${\mathcal{M}}_{qc}(\sigma)$$ of quadratic line complexes with a fixed Segre symbol σ as well as the moduli space $${\mathcal{M}}_{ss}(\sigma)$$ of the
On the Integrability of (2+1)-Dimensional Quasilinear Systems
A (2+1)-dimensional quasilinear system is said to be ‘integrable’ if it can be decoupled in infinitely many ways into a pair of compatible n-component one-dimensional systems in Riemann invariants.
Integrable systems and modular forms of level 2
A set of nonlinear differential equations associated with the Eisenstein series of the congruent subgroup $\Gamma_0(2)$ of the modular group $SL_2(\mathbb{Z})$ is constructed. These nonlinear
Nonlinear wave equation, nonlinear Riemann problem, and the twistor transform of Veronese webs
Veronese webs are rich geometric structures with deep relationships to various domains of mathematics. The PDEs which determine the Veronese web are overdetermined if dim >3, but in the case dim =3
SOLITON EIGENFUNCTION EQUATIONS: THE IST INTEGRABILITY AND SOME PROPERTIES
Eigenfunctions of the linear eigenvalue problems for the soliton equations obey nonlinear differential equations. It is shown that these eigenfunction equations are integrable by the inverse spectral
...
...