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Ufimsk. Mat. Zh., 2020, Volume 12, Issue 2, Pages 49–54 (Mi ufa518)  

On equivalence of one spin system and two-component Camassa-Holm equation

A. G. Tayshievaa, T. R. Myrzakulb, G. N. Nugmanovaa

a L.N. Gumilyov Eurasian National University, Satpayev str. 2, Z05T8G1, Nur-Sultan, Kazakhstan
b Kazakh State Women's Teacher Training University, Aiteke bi str. 99, A15A4G6, Almaty, Kazakhstan

Abstract: The work is devoted to studying an equivalence of a two-component Camassa-Holm equation and a spin system being a generalization of Heisenberg ferromagnet equation. It is known that the equivalence between two nonlinear integrable equations provides a possibility of an extended search of their various exact solutions. For Camassa-Holm equation, a method of inverse scattering problem can be applied via a system of linear partial differential equations with scalar coefficients. Contrary to Camassa-Holm equation, the coefficients of linear system corresponding to spin equations are related with symmetric matrix Lax representations. This is why, while establishing an equivalence between two above equations, additional difficulties arise. In view of this, we propose a matrix Lax representation for Camassa-Holm equation in a symmetric space. Employing this result, we establish a gauge equivalence between two-component Camassa-Holm equation and a spin system. We describe a relation between their solutions.

Keywords: two-component Camassa-Holm equation, matrix Lax representation, spin system, gauge equivalence.

Funding Agency Grant Number
Ministry of Education and Science of the Republic of Kazakhstan 132
The work is financially supported by Ministry of Education and Science of the Republic of Kazakhstan (contract no. 132 on 12.03.2018).


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English version:
Ufa Mathematical Journal, 2020, 12:2, 50–55 (PDF, 290 kB); https://doi.org/10.13108/2020-12-2-50

Bibliographic databases:

UDC: 517.951, 517.957
MSC: 35C08, 35Q51
Received: 10.12.2019

Citation: A. G. Tayshieva, T. R. Myrzakul, G. N. Nugmanova, “On equivalence of one spin system and two-component Camassa-Holm equation”, Ufimsk. Mat. Zh., 12:2 (2020), 49–54; Ufa Math. J., 12:2 (2020), 50–55

Citation in format AMSBIB
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\by A.~G.~Tayshieva, T.~R.~Myrzakul, G.~N.~Nugmanova
\paper On equivalence of one spin system and two-component Camassa-Holm equation
\jour Ufimsk. Mat. Zh.
\yr 2020
\vol 12
\issue 2
\pages 49--54
\mathnet{http://mi.mathnet.ru/ufa518}
\transl
\jour Ufa Math. J.
\yr 2020
\vol 12
\issue 2
\pages 50--55
\crossref{https://doi.org/10.13108/2020-12-2-50}
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