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Tr. Mat. Inst. Steklova, 2018, Volume 302, Pages 268–286 (Mi tm3932)  

Darboux system: Liouville reduction and an explicit solution

R. Ch. Kulaevab, A. K. Pogrebkovcd, A. B. Shabatef

a North-Ossetian State University named after K. L. Khetagurov, ul. Vatutina 44–46, Vladikavkaz, 362025 Russia
b Southern Mathematical Institute – the Affiliate of Vladikavkaz Scientific Centre of Russian Academy of Sciences, ul. Vatutina 53, Vladikavkaz, 362027 Russia
c Steklov Mathematical Institute of Russian Academy of Sciences, ul. Gubkina 8, Moscow, 119991 Russia
d National Research University "Higher School of Economics", ul. Myasnitskaya 20, Moscow, 101000 Russia
e Karachay-Cherkess State University named after U. D. Aliyev, ul. Lenina 29, Karachaevsk, 369202 Russia
f L.D. Landau Institute for Theoretical Physics of Russian Academy of Sciences, pr. Akademika Semenova 1a, Chernogolovka, Moscow oblast, 142432 Russia

Abstract: A class of solutions to a Darboux system in $\mathbb R^3$ is introduced that satisfy the factorization condition for an auxiliary second-order linear problem. It is shown that this reduction provides the (local) solvability of the Darboux system, and an explicit solution is given to this problem for two types of dependent variables. Explicit formulas for the Lamé coefficients and solutions to the associated linear problem are constructed. It is shown that the reduction, known in the literature, to a weakly nonlinear system is a particular case of the approach proposed.

Funding Agency Grant Number
Russian Academy of Sciences - Federal Agency for Scientific Organizations PRAS-18-02
Ministry of Education and Science of the Russian Federation
This work was supported in part by the Program of the Presidium of the Russian Academy of Sciences no. 02 “Nonlinear Dynamics: Fundamental Problems and Applications.” The second author was also supported in part by the Russian Academic Excellence Project “5-100.”


DOI: https://doi.org/10.1134/S0371968518030123

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English version:
Proceedings of the Steklov Institute of Mathematics, 2018, 302, 250–269

Bibliographic databases:

Document Type: Article
UDC: 517.953+517.956.35+517.957
Received: March 23, 2018

Citation: R. Ch. Kulaev, A. K. Pogrebkov, A. B. Shabat, “Darboux system: Liouville reduction and an explicit solution”, Topology and physics, Collected papers. Dedicated to Academician Sergei Petrovich Novikov on the occasion of his 80th birthday, Tr. Mat. Inst. Steklova, 302, MAIK Nauka/Interperiodica, Moscow, 2018, 268–286; Proc. Steklov Inst. Math., 302 (2018), 250–269

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\pages 268--286
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