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
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.
two-component Camassa-Holm equation, matrix Lax representation, spin system, gauge equivalence.
PDF file (375 kB)
Ufa Mathematical Journal, 2020, 12:2, 50–55 (PDF, 290 kB); https://doi.org/10.13108/2020-12-2-50
MSC: 35C08, 35Q51
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
\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.
\jour Ufa Math. J.
Citing articles on Google Scholar:
Related articles on Google Scholar:
|Number of views:|