Finding solutions of parabolic Monge-Ampere equations by using the geometry of sections of the contact distribution

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Authors

ALEKSEEVSKIY Dmitry MANNO Gianni PUGLIESE Fabrizio ALONCO-BLANCO Ricardo

Year of publication 2014
Type Article in Periodical
Magazine / Source Differential Geometry and its Applications
MU Faculty or unit

Faculty of Science

Citation
Web https://doi.org/10.1016/j.difgeo.2013.10.015
Doi http://dx.doi.org/10.1016/j.difgeo.2013.10.015
Field General mathematics
Keywords Parabolic Monge Ampere equations; Characteristic distribution; Construction of solutions
Description In a series of papers we have described normal forms of parabolic Monge–Ampere equations (PMAEs) by means of their characteristic distribution. In particular, PMAEs with two independent variables are associated with Lagrangian (or Legendrian) subdistributions of the contact distribution of a 5-dimensional contact manifold. The geometry of sections of the contact distribution allowed us to get the aforementioned normal forms. In the present work, for a distinguished class of PMAEs, we will construct 3-parametric families of solutions starting from particular sections of the characteristic distribution. We will illustrate the method by several concrete computations. Moreover, we will see, for some linear PMAEs, how to construct a recursive process for obtaining new solutions. At the end, after showing that some classical equations on affine connected 3-dimensional manifolds are PMAEs, we will apply the integration method to some particular examples.
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