On the integrability of symplectic Monge–Ampère equations

@article{Doubrov2010OnTI,
  title={On the integrability of symplectic Monge–Amp{\`e}re equations},
  author={Boris Doubrov and Eugene V. Ferapontov},
  journal={Journal of Geometry and Physics},
  year={2010},
  volume={60},
  pages={1604-1616}
}

The moment map on the space of symplectic 3D Monge-Amp\`ere equations

For any 2nd order scalar PDE E in one unknown function, that we interpret as a hypersurface of a 2nd order jet space J2, we construct, by means of the characteristics of E, a sub–bundle of the

Integrability of Dispersionless Hirota-Type Equations and the Symplectic Monge–Ampère Property

We prove that integrability of a dispersionless Hirota-type equation implies the symplectic Monge–Ampère property in any dimension $\geq 4$. In 4D, this yields a complete classification of

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

Contact geometry of multidimensional Monge-Ampere equations: characteristics, intermediate integrals and solutions

We study the geometry of multidimensional scalar 2 order PDEs (i.e. PDEs with n independent variables) with one unknown function, viewed as hypersurfaces E in the Lagrangian Grassmann bundle M (1)

Lax pairs, recursion operators and bi-Hamiltonian representations of (3+1)-dimensional Hirota type equations

The integrable heavenly type equations and their Lie-algebraic structure

There are investigated the Lie algebraic structure and integrability properties of a very interesting class of nonlinear dynamical systems called the heavenly equations, which were initiated by

Conformal geometry of surfaces in the Lagrangian Grassmannian and second‐order PDE

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 Möbius space S1,2. Using Cartan's

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
...

References

SHOWING 1-10 OF 48 REFERENCES

A classification of Monge-Ampère equations

A classical problem of a local classification of non-linear equation arising in S. Lie works is studied for the most natural class of Monge-Ampere equations (M.A.E.) on a smooth manifold M". We solve

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

Hydrodynamic reductions of multidimensional dispersionless PDEs: The test for integrability

A (d+1)-dimensional dispersionless PDE is said to be integrable if its n-component hydrodynamic reductions are locally parametrized by (d−1)n arbitrary functions of one variable. The most important

Hyper-Kähler Hierarchies and Their Twistor Theory

Abstract: A twistor construction of the hierarchy associated with the hyper-Kähler equations on a metric (the anti-self-dual Einstein vacuum equations, ASDVE, in four dimensions) is given. The

Contact geometry and non-linear differential equations

Introduction Part I. Symmetries and Integrals: 1. Distributions 2. Ordinary differential equations 3. Model differential equations and Lie superposition principle Part II. Symplectic Algebra: 4.

On symplectic classification of effective 3-forms and Monge–Ampère equations

On the solutions of the second heavenly and Pavlov equations

We have recently solved the inverse scattering problem for one-parameter families of vector fields, and used this result to construct the formal solution of the Cauchy problem for a class of

On classification of second-order PDEs possessing partner symmetries

Recently we have demonstrated how to use partner symmetries for obtaining noninvariant solutions of the heavenly equations of Plebañski that govern heavenly gravitational metrics. In this paper, we

On self-dual gravity.

  • Grant
  • Mathematics
    Physical review. D, Particles and fields
  • 1993
The Ashtekar-Jacobson-Smolin equations are studied and it is found that any self-dual metric can be characterised by a function that satisfies a nonlinear evolution equation, to which the general solution can be found iteratively.

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.