An introduction to completely exceptional second order scalar partial differential equations

Moreno, Giovanni

doi:10.4064/bc113-0-15

Cited by 3 publications

(4 citation statements)

References 12 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The key feature of (108) is that it is not Cont(M )-invariant. Making (108) into a Cont(M )-invariant test is not an easy task, and the heavy machinery used in [23] confirms that; see also [40]. Nevertheless, the result is surprisingly simple, and even easy to guess.…”

Section: 5mentioning

confidence: 77%

See 1 more Smart Citation

Geometry of Lagrangian Grassmannians and nonlinear PDEs

Gutt¹,

Manno

Moreno

2019

Banach Center Publ.

Self Cite

View full text Add to dashboard Cite

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 hypersurfaces in the Lagrangian Grassmannian and second-order PDEs. Equipped with a comprehensive bibliography, this paper has been especially designed as an opening contribution for the proceedings volume of the homonymous workshop held in Warsaw, September 5-9, 2016, and organised by the authors.

show abstract

Section: 5mentioning

confidence: 77%

“…encompasses the transformations of the form ( 38) and (40). Those of the form (39), that is S 2 V * acting by translations on the big cell S 2 V * , clearly allow us to move the origin arctan(0) into any other point arctan(h) of the big cell.…”

Section: Natural Group Actions Onmentioning

confidence: 99%

Geometry of Lagrangian Grassmannians and nonlinear PDEs

Gutt¹,

Manno

Moreno

2019

Banach Center Publ.

Self Cite

View full text Add to dashboard Cite

show abstract

“…has rank n + κ, which in turn is equivalent to the upper-right block having rank κ: but, in view of (32), this is the same as the rank of the matrix (31).…”

Section: The Contact Cone Structure Associated With a Second-order Pdementioning

confidence: 99%

“…and a straightforward computation shows that the rank of their symbols is less or equal to 2: as such, these PDEs are a proper subclass of a larger class of Monge-Ampère equations that were introduced later by Boillat in [6], as the only PDEs whose characteristic velocities behave in the "completely exceptional" way in the sense of P. Lax. [7,8,21,27,28,31].…”

mentioning

confidence: 99%

The moment map on the space of symplectic 3D Monge-Ampère equations

Gutt¹,

Manno²,

Moreno³

et al. 2021

Preprint

Self Cite

View full text Add to dashboard Cite

For any 2 nd order scalar PDE E in one unknown function, that we interpret as a hypersurface of a secondorder jet space J 2 , we construct, by means of the characteristics of E, a sub-bundle of the contact distribution of the underlying contact manifold J 1 , consisting of conic varieties. We call it the contact cone structure associated with E. We then focus on symplectic Monge-Ampère equations in 3 independent variables, that are naturally parametrized by a 13-dimensional real projective space. If we pass to the field of complex numbers C, this projective space turns out to be the projectivization of the 14-dimensional irreducible representation of the simple Lie group Sp(6, C): the associated moment map allows to define a rational map from the space of symplectic 3D Monge-Ampère equations to the projectivization of the space of quadratic forms on a 6-dimensional symplectic vector space. We study in details the relationship between the zero locus of the image of , herewith called the cocharacteristic variety, and the contact cone structure of a 3D Monge-Ampère equation E: under the hypothesis of non-degenerate symbol, we prove that these two constructions coincide. A key tool in achieving such a result will be a complete list of mutually non-equivalent quadratic forms on a 6-dimensional symplectic space, which has an interest on its own. Contents 1. Introduction 1 2. Preliminaries 3 3. The contact cone structure associated with a second-order PDE 7 4. Quadric contact cone structures associated with 3D Monge-Ampère equations 10 5. Reconstructing a second-order PDE from a contact cone structure 13 6. The space PΛ 3 0 (C) of symplectic 3D Monge-Ampère equations 15 7. Normal forms of quadratic forms on C with respect to Sp(C) 18 8. The moment map on the space of Monge-Ampère equations 25 9. Hyperplane sections of the Lagrangian Grassmannian LGr(3, C) 30 10. Conclusions and perspectives 35 References 35

show abstract

On some exact solutions of heavenly equations in four dimensions

Stȩpień

2020

AIP Advances

View full text Add to dashboard Cite

Some new classes of exact solutions (so-called functionally invariant solutions) of the elliptic and hyperbolic complex Monge–Ampère equations and of the second heavenly equation are found. Besides, non-invariance of the found classes of solutions of the hyperbolic complex Monge–Ampère equation and the second heavenly equation is investigated—these classes of solutions determine the new classes of metrics without Killing vectors. A criterion of non-invariance of the solutions belonging to the found classes is also formulated.

show abstract

An introduction to completely exceptional second order scalar partial differential equations

Cited by 3 publications

References 12 publications

Geometry of Lagrangian Grassmannians and nonlinear PDEs

Geometry of Lagrangian Grassmannians and nonlinear PDEs

The moment map on the space of symplectic 3D Monge-Ampère equations

On some exact solutions of heavenly equations in four dimensions

Contact Info

Product

Resources

About