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June 2019, 11(2): 167-185. doi: 10.3934/jgm.2019009
Riemann-Hilbert problem, integrability and reductions
Vladimir S. Gerdjikov 1,2,5, , Rossen I. Ivanov 3,, and Aleksander A. Stefanov 2,4,
Department of Applied Mathematics, National Research Nuclear University MEPHI, 31 Kashirskoe Shosse, Moscow 115409, Russian Federation
Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, 8 Acad. G. Bonchev Street, Sofia 1113, Bulgaria
School of Mathematical Sciences, Technological University Dublin - City Campus, Kevin Street, Dublin D08 NF82, Ireland
Faculty of Mathematics and Infromatics, Sofia University St. Kliment Ohridsky, 5 James Bourchier Blvd., Sofia 1164, Bulgaria
Institute for Advanced Physical Studies, New Bulgarian University, 21 Montevideo Street, Sofia 1618, Bulgaria
* Corresponding author: R. I. Ivanov
Received April 2018 Revised March 2019 Published May 2019
Figure(3)
The present paper is dedicated to integrable models with Mikhailov reduction groups $G_R \simeq \mathbb{D}_h.$ Their Lax representation allows us to prove, that their solution is equivalent to solving Riemann-Hilbert problems, whose contours depend on the realization of the $G_R$-action on the spectral parameter. Two new examples of Nonlinear Evolution Equations (NLEE) with $\mathbb{D}_h$ symmetries are presented.
Keywords: Inverse scattering, semisimple Lie algebras, solitons.
Mathematics Subject Classification: Primary: 35K10, 35Q15; Secondary: 37K15, 35Q55.
Citation: Vladimir S. Gerdjikov, Rossen I. Ivanov, Aleksander A. Stefanov. Riemann-Hilbert problem, integrability and reductions. Journal of Geometric Mechanics, 2019, 11 (2) : 167-185. doi: 10.3934/jgm.2019009
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Figure 1. Contour of a RHP with $ \mathbb{Z}_3 $ symmetry
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Figure 2. Contour of the RHP $ \mathbb{D}_3 $ symmetry
Figure 3. Contour of the RHP for $ \mathbb{D}_2 $ symmetry (upper panel) and for $ \mathbb{D}_4 $ symmetry (lower panel)
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Vladimir S. Gerdjikov Rossen I. Ivanov Aleksander A. Stefanov