Geodesic
Simple solutions of three equations of mathematical physics
Trudy Moskovskogo matematičeskogo obŝestva, Trudy Moskovskogo Matematicheskogo Obshchestva, Tome 79 (2018) no. 2, pp. 221-236.

Voir la notice de l'article provenant de la source Math-Net.Ru

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). 
@article{MMO_2018_79_2_a2,
     author = {V. K. Beloshapka},
     title = {Simple solutions of three equations of mathematical physics},
     journal = {Trudy Moskovskogo matemati\v{c}eskogo ob\^{s}estva},
     pages = {221--236},
     publisher = {mathdoc},
     volume = {79},
     number = {2},
     year = {2018},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MMO_2018_79_2_a2/}
}
TY  - JOUR
AU  - V. K. Beloshapka
TI  - Simple solutions of three equations of mathematical physics
JO  - Trudy Moskovskogo matematičeskogo obŝestva
PY  - 2018
SP  - 221
EP  - 236
VL  - 79
IS  - 2
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/MMO_2018_79_2_a2/
LA  - ru
ID  - MMO_2018_79_2_a2
ER  - 
%0 Journal Article
%A V. K. Beloshapka
%T Simple solutions of three equations of mathematical physics
%J Trudy Moskovskogo matematičeskogo obŝestva
%D 2018
%P 221-236
%V 79
%N 2
%I mathdoc
%U http://geodesic.mathdoc.fr/item/MMO_2018_79_2_a2/
%G ru
%F MMO_2018_79_2_a2
V. K. Beloshapka. Simple solutions of three equations of mathematical physics. Trudy Moskovskogo matematičeskogo obŝestva, Trudy Moskovskogo Matematicheskogo Obshchestva, Tome 79 (2018) no. 2, pp. 221-236. http://geodesic.mathdoc.fr/item/MMO_2018_79_2_a2/

[1] A. Ostrowski, “Über Dirichletsche Reihen und algebraische Differentialgleichungen”, Math. Z., 8 (1920), 241–298 | DOI | MR | Zbl

[2] A. G. Vitushkin, “13-ya problema Gilberta i smezhnye voprosy”, UMN, 59:1(355) (2004), 11–24 | DOI

[3] V. K. Beloshapka, “Analytic Complexity of Functions of Two Variables”, Russian Journal of Mathematical Physics, 14:3 (2007), 243–249 | DOI | MR | Zbl

[4] V. K. Beloshapka, “Stabilizer of a Function in the Gauge Group”, Russian Journal of Mathematical Physics, 24:2 (2017), 148–152 | DOI | MR | Zbl

[5] V. K. Beloshapka, “Analytic complexity of functions of two variables”, Russ. J. Math. Phys., 14:3 (2007), 243–249 | DOI | MR | Zbl

[6] A. V. Domrin, “Meromorfnoe prodolzhenie reshenii solitonnykh uravnenii”, Izv. RAN. Ser. matem., 74:3 (2010), 23–44 | DOI | Zbl

[7] J. Liouville, “Sur l'equation aux differences partielles $\frac{\partial^2\ln\lambda}{\partial u \partial v}\pm 2\lambda q^2=0$”, J. Math. Pure Appl., 18 (1853), 71–72