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ISSN 1817-2172, рег. Эл. № ФС77-39410, ВАК

Differential Equations and Control Processes
(Differencialnie Uravnenia i Protsesy Upravlenia)

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The Integration of Harry Dym and Korteweg-de Vries Equations in Parametric Form

Author(s):

Natalya Konstantinovna Volosova

4th year student
Dept. of applied mathematics 1
Moscow State University of Railway
Engineering, 9, of. 9, St. Obraztsova, Moscow, 127994, Russia

konstantinvolosov@yandex.ru

Aleksandra Konstantinovna Volosova

konstantinvolosov@yandex.ru

Konstantin Aleksandrovich Volosov

DSc in phisics and mathematics
Professor
Dept. of applied mathematics 1
Moscow State University of Railway
Engineering, 9, of. 9, St. Obraztsova, Moscow, 127994, Russia

konstantinvolosov@yandex.ru

Abstract:

In the present work we for the first time apply a relatively new method of constructive unfixed change of variables to the Harry Dym (HD) and the Korteweg--de Vries (KdV) equations. We construct two dynamical systems and formulate necessary conditions for the stability of phase trajectories. A system of functional algebraic equations is constructed and it is proved that two formal solvability conditions for a system of first order partial differential equations have one non--trivial common factor. An important feature of the HD and KdV equations was found: after an unfixed constructive change of variables, a new "hidden" key equation for the function of the partial first derivative can be separated from the other equations. The exact solutions constructed with the help of a non-autonomous dynamical system coincide with global solutions. That is not the case for equations with dissipation. Two classes of exact solutions are found for the HD and for the KdV equation. A possibility arises to construct new asymptotic solutions.

Keywords

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