Classification of Monge–Ampeère Equations

Dispersionless integrable systems in 3D and Einstein-Weyl geometry

“…• Generalized Laplace invariants of PDEs appearing in the context of Darboux integrability can be obtained from the Laplace invariants of linearized equations [69,2,42,45,44].…”

Geometry of Lagrangian Grassmannians and nonlinear PDEs

“…Due to the invariance of the framework, the sub-distribution X E of C can by all means replace E in the treatment of the equivalence problem. This point of view is at the basis of many works about invariants and classification of Goursat-type Monge-Ampère equations, see, e.g., [4,16,9,13,34,33].…”

“…It is not hard to realise that, after the substitutions the equation (32) becomes (33). On the top of that, the hyperplane π H , with coefficients given by (34) is tangent to LGr (3,6). This is not hard to see: the left-hand side of (33), regarded as a function of h, vanishes at h = H, together with its first derivatives.…”

An introduction to completely exceptional second order scalar partial differential equations

“…This explains the codimension one. The definition of Monge-Ampère equations as hypersurfaces of the form E ω was given in 1978 by V. Lychagin [11]. Alternatively, Monge-Ampère equations can be defined as hyperplane sections of the fibres of M (1) .…”

“…It is somewhat useful to refer to the integral submanifolds U of the contact EDS (M, θ) as candidate solutions. Indeed, thanks to (11), candidate solutions are in (a local) one-to-one correspondence with functions in n variables. A candidate solution may be thought of as a solution of the trivial equation 0 = 0; it becomes a solution of the (nontrivial) equation ( 6) only if it is contained into E. In a sense, the whole machinery so far introduced just allowed to rephrase in terms of a set-theoretical inclusion the property for a function and its 1 st derivatives to satisfy a certain relation.…”