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Chaplygin and Keldysh normal forms of Monge-Ampère equations of variable type. (English. Russian original) Zbl 0799.35164

Math. Notes 52, No. 5, 1121-1124 (1992); translation from Mat. Zametki 52, No. 5, 63-67 (1992).
We give necessary and sufficient conditions for the local equivalence of Monge-Ampère equations of variable type with smooth coefficients to equations of the form \[ \begin{aligned} q_ 1^ n &{{\partial^ 2 u} \over {\partial q_ 2^ 2}} + {{\partial^ 2 u} \over {\partial q_ 1^ 2}} +\alpha {{\partial u} \over {\partial q_ 1}} +\beta {{\partial u} \over {\partial q_ 2}} =0, \tag{1}\\ q_ 1^ n &{{\partial^ 2u} \over {\partial q_ 1^ 2}} + {{\partial^ 2 u} \over {\partial q_ 2^ 2}} +\alpha {{\partial u} \over {\partial q_ 1}} +\beta {{\partial u} \over {\partial q_ 2}} =0.\tag{2}\end{aligned} \] Here \(n\in \mathbb{N}\); \(\alpha,\beta\in \mathbb{R}\). The special form of (1) for \(\alpha= \beta=0\) is known as Chaplygin’s equation and (2) as Keldysh’s equation. The solution of the equivalence problem is based on the connection between differential forms on the manifold of jets and Monge-Ampère equations [V. V. Lycagin, Usp. Mat. Nauk 34, No. 1(205), 137-165 (1979; Zbl 0405.58003)].

MSC:

35M10 PDEs of mixed type
35A30 Geometric theory, characteristics, transformations in context of PDEs
37J99 Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems
37C80 Symmetries, equivariant dynamical systems (MSC2010)

Citations:

Zbl 0405.58003
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References:

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