Related papers: Contact geometry of multidimensional Monge-Amp\`er…
In this paper we review a geometric approach to PDEs. We mainly focus on scalar PDEs in n independent variables and one dependent variable of order one and two, by insisting on the underlying (2n+1)-dimensional contact manifold and the…
This paper is a natural companion of [Alekseevsky D.V., Alonso Blanco R., Manno G., Pugliese F., Ann. Inst. Fourier (Grenoble) 62 (2012), 497-524, arXiv:1003.5177], generalising its perspectives and results to the context of third-order…
We study the contact geometry of scalar second order hyperbolic equations in the plane of generic type. Following a derivation of parametrized contact-invariants to distinguish Monge-Ampere (class 6-6), Goursat (class 6-7) and generic…
The classes of Monge-Amp\`ere systems, decomposable and bi-decomposable Monge-Amp\`ere systems, including equations for improper affine spheres and hypersurfaces of constant Gauss-Kronecker curvature are introduced. They are studied by the…
We describe some natural relations connecting contact geometry, classical Monge-Ampere equations and theory of singularities of solutions to nonlinear PDEs. They reveal the hidden meaning of Monge-Ampere equations and sheds new light on…
Using Monge-Amp\`ere geometry, we study the singular structure of a class of nonlinear Monge-Amp\`ere equations in three dimensions, arising in geophysical fluid dynamics. We extend seminal earlier work on Monge-Amp\`ere geometry by…
We develop a method for describing invariant Monge-Amp\`ere equations in the sense of V. Lychagin and T. Morimoto (MAE) on a homogeneous contact manifold $N$ of a semisimple Lie group $G$, which is the contactification of the homogeneous…
This paper contains a thorough introduction to the basic geometric properties of the manifold of Lagrangian subspaces of a linear symplectic space, known as the Lagrangian Grassmannian. It also reviews the important relationship between…
We use methods from exterior differential systems (EDS) to develop a geometric theory of scalar, first-order Lagrangian functionals and their associated Euler-Lagrange PDEs, subject to contact transformations. The first chapter contains an…
k-Contact geometry is a generalisation of contact geometry to analyse field theories. We develop an approach to k-contact geometry based on distributions that are distributionally maximally non-integrable and admit, locally, k commuting…
Of all real Lagrangian--Grassmannians $LG(n,2n)$, only $LG(2,4)$ admits a distinguished (Lorentzian) conformal structure and hence is identified with the indefinite M\"obius space $S^{1,2}$. Using Cartan's method of moving frames, we study…
A PhD thesis written under supervision of Pawel Nurowski and defended at the Faculty of Physics of the University of Warsaw. We adress the problems of local equivalence and geometry of third order ODEs modulo contact, point and…
By studying the development of shock waves out of discontinuity waves, in 1954 P. Lax discovered a class of PDEs, which he called 'completely exceptional', where such a transition does not occur after a finite time. A straightforward…
Let V be a real hypersurface of class C^k, k>=3, in a complex manifold M of complex dimension n+1, HT(V) the holomorphic tangent bundle to V giving the induced CR structure on V. Let \theta be a contact form for (V,HT(V)), \xi_0 the Reeb…
We find normal forms for parabolic Monge-Ampere equations. Of these, the most general one holds for any equation admitting a complete integral. Moreover, we explicitly give the determining equation for such integrals; restricted to the…
As an application of the general theory on extrinsic geometry, we investigate extrinsic geometry in frag varieties and systems of linear PDE's for a class of special interest associated with the adjoint representation of $\mathfrak{sl}(3)$.…
In this paper, we study symmetry and existence of solutions of minimal gradient graph equations on punctured space $\mathbb R^n\setminus\{0\}$, which include the Monge-Amp\`ere equation, inverse harmonic Hessian equation and the special…
In this paper we consider symplectic and contact Lie algebras. We define contactization and symplectization procedures and describe its main properties. We also give classification of such algebras in dimensions 3 and 4. The classification…
For any second-order scalar PDE $\mathcal{E}$ in one unknown function, that we interpret as a hypersurface of a second-order jet space $J^2$, we construct, by means of the characteristics of $\mathcal{E}$, a sub-bundle of the contact…
We address the problem of local geometry of third order ODEs modulo contact, point and fibre-preserving transformations of variables. Several new and already known geometries are described in a uniform manner by the Cartan method of…