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 Tr. Mosk. Mat. Obs., 2018, Volume 79, Issue 2, Pages 221–236 (Mi mmo613)

Simple solutions of three equations of mathematical physics

V. K. Beloshapka

Department of Mechanics and Mathematics, Lomonosov Moscow State University

Abstract: In this paper, we consider three equations of mathematical physics for functions of two variables: the heat equation, the Liouville equation, and the Korteweg-de Vries (KdV) equation. We obtain complete lists of simple solutions for all three equations, that is, solutions of analytic complexity not exceeding one. All solutions of this type for the heat equation can be expressed in terms of the error function (Theorem 1) and form a 4-parameter family; for the Liouville equation, the answer is the union of a 6-parameter family and a 3-parameter family of elementary functions (Theorem 2); for the Korteweg-de Vries equation, the list consists of four 3-parameter families containing elementary and elliptic functions (Theorem 3).

Key words and phrases: analytic functions, analytic complexity, differential polynomials, equations of mathematical physics.

 Funding Agency Grant Number Russian Foundation for Basic Research 17-01-00592_à

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English version:
Transactions of the Moscow Mathematical Society, 2018, 187–200

UDC: 517.55, 517.923, 514.74
MSC: 32A, 33C, 35A24
Revised: 19.02.2018

Citation: V. K. Beloshapka, “Simple solutions of three equations of mathematical physics”, Tr. Mosk. Mat. Obs., 79, no. 2, MCCME, M., 2018, 221–236; Trans. Moscow Math. Soc., 2018, 187–200

Citation in format AMSBIB
\Bibitem{Bel18} \by V.~K.~Beloshapka \paper Simple solutions of three equations of mathematical physics \serial Tr. Mosk. Mat. Obs. \yr 2018 \vol 79 \issue 2 \pages 221--236 \publ MCCME \publaddr M. \mathnet{http://mi.mathnet.ru/mmo613} \elib{http://elibrary.ru/item.asp?id=37045098} \transl \jour Trans. Moscow Math. Soc. \yr 2018 \pages 187--200 \crossref{https://doi.org/10.1090/mosc/280} \scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-85060845671}