Simple solutions of three equations of mathematical physics

Beloshapka, V.

doi:10.1090/mosc/280

Simple solutions of three equations of mathematical physics
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by V. K. Beloshapka
Translated by: Alexander Shtern

Trans. Moscow Math. Soc. 2018, 187-200

DOI: https://doi.org/10.1090/mosc/280

Published electronically: November 29, 2018

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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).

References

Alexander Ostrowski, Über Dirichletsche Reihen und algebraische Differentialgleichungen, Math. Z. 8 (1920), no. 3-4, 241–298 (German). MR 1544442, DOI 10.1007/BF01206530
A. G. Vitushkin, Hilbert’s thirteenth problem and related questions, Uspekhi Mat. Nauk 59 (2004), no. 1(355), 11–24 (Russian, with Russian summary); English transl., Russian Math. Surveys 59 (2004), no. 1, 11–25. MR 2068840, DOI 10.1070/RM2004v059n01ABEH000698
V. K. Beloshapka, Analytic complexity of functions of two variables, Russ. J. Math. Phys. 14 (2007), no. 3, 243–249. MR 2341772, DOI 10.1134/S1061920807030016
V. K. Beloshapka, Stabilizer of a function in the Gage group, Russ. J. Math. Phys. 24 (2017), no. 2, 148–152. MR 3658407, DOI 10.1134/S1061920817020029
A. V. Domrin, Meromorphic extension of solutions of soliton equations, Izv. Ross. Akad. Nauk Ser. Mat. 74 (2010), no. 3, 23–44 (Russian, with Russian summary); English transl., Izv. Math. 74 (2010), no. 3, 461–480. MR 2682370, DOI 10.1070/IM2010v074n03ABEH002494
J. Liouville, Sur l’équation 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.