Exact linearization of fully nonlinear Monge – Ampère type equations
https://doi.org/10.26583/vestnik.2025.6.5
EDN: KREEFH
Abstract
New classes of Monge – Ampère equations of a fairly general form are described, depending on one to six arbitrary functions of one or two arguments that allow exact linearization in closed form. For linearization, contact Euler and Legendre transformations and special point transformations (including the nonclassical hodograph transformation) of their combinations are used. Special attention is given to the Monge – Ampère equations encountered in meteorology and geophysics. Equivalence transformations of classes of Monge – Ampère equations of a special kind are also considered. For some nonlinear equations, exact solutions were obtained depending on arbitrary functions. Two nonstationary, strongly nonlinear Monge-Ampère type equations with three independent variables, encountered in electron magnetohydrodynamics and geophysical fluid dynamics, were also considered. For these equations, two-dimensional reductions to simpler equations that allow exact linearization were constructed in traveling-wave variables.
Keywords
Monge – Ampère equations,
fully nonlinear PDEs,
exact linearization,
Euler and Legendre transformations,
hodograph transformation,
contact and point transformations
Supporting agencies: The work was carried out on the topics of the state assignment (state registration numbers 124012500440-9 and FSWU-2023-0031
About the Authors
A. D. Polyanin
Ishlinsky Institute for Problems in Mechanics, Russian Academy of Sciences
Russian Federation
Russian Federation
A. V. Aksenov
Lomonosov Moscow State University
Russian Federation
Russian Federation
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For citations:
Polyanin A.D., Aksenov A.V. Exact linearization of fully nonlinear Monge – Ampère type equations. Vestnik natsional'nogo issledovatel'skogo yadernogo universiteta "MIFI". 2025;14(6):500-515. (In Russ.) https://doi.org/10.26583/vestnik.2025.6.5. EDN: KREEFH
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