Seminar talk, 7 June 2023

From Geometry of Differential Equations
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Speaker: Svetlana Mukhina

Title: Contact vs symplectic geometry

The report will show how some symplectic Monge-Ampère type equations can be solved by applying contact transformations to them.

As is known, symplectic Monge-Ampère equations with two independent variables are locally symplectic equivalent to linear equations with constant coefficients if and only if the corresponding Nijenhuis bracket is zero (the Lychagin-Rubtsov theorem). Necessary and sufficient conditions for the contact equivalence of the general (not necessarily symplectic) Monge-Ampère linear equations were found by Kushner.

Using these results, we consider the problem of constructing exact solutions to some equations arising in filtration theory. We will consider a model of unsteady displacement of oil by a solution of active reagents. This model describes the process of oil extraction from hard-to-recover deposits. This model is described by a hyperbolic system of partial differential equations of the first order of the Jacobi type. Unknown functions are the water saturation and concentration of reagents in an aqueous solution, and independent variables are time and linear coordinate.

With the help of symplectic and contact transformations, it is possible to reduce the model equations to a linear wave equation. The exact solution of this system is obtained and the Cauchy problem is solved.