Analytical solutions and reductions of the nonlinear equation of geophysical hydrodynamics
https://doi.org/10.26583/vestnik.2026.1.6
EDN: ROURGW
Abstract
A strongly nonlinear, time-dependent Monge–Ampère type equation with three independent variables, encountered in geophysical fluid dynamics, is investigated. A number of exact analytical solutions with additive, multiplicative, generalized, and functional separation of variables are obtained. Special attention is given to the construction of exact closed-form solutions that are expressed in terms of elementary functions. Two-dimensional reductions leading to simpler partial differential equations with two independent variables (including stationary Monge – Ampère equations, the linear wave equation, etc.) are considered. Some one-dimensional reductions are described that allow obtaining solutions satisfying ordinary differential equations or systems of such equations. The obtained results and exact solutions can be used to evaluate the accuracy and analyze the adequacy of numerical and approximate analytical methods for solving problems described by strongly nonlinear partial differential equations.
Keywords
Monge – Ampère type equations,
fully nonlinear PDEs,
geophysical fluid dynamics,
exact analytical solutions,
two- and one-dimensional reductions
Supporting agencies: The work was completed on the topics of the state assignment (state registration numbers 124012500440-9 and FSWU-2023-0031).
About the Authors
A. V. Aksenov
Lomonosov Moscow State University
Russian Federation
Russian Federation
A. D. Polyanin
Ishlinsky Institute for Problems in Mechanics, Russian Academy of Sciences
Russian Federation
Russian Federation
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For citations:
Aksenov A.V., Polyanin A.D. Analytical solutions and reductions of the nonlinear equation of geophysical hydrodynamics. Vestnik natsional'nogo issledovatel'skogo yadernogo universiteta "MIFI". 2026;15(1):62-79. (In Russ.) https://doi.org/10.26583/vestnik.2026.1.6. EDN: ROURGW
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