arXiv subjects

Akbota Myrzakul

Publications and source records attributed to Akbota Myrzakul.

4 recordsLinked to original sources

arXiv2018

Notes on Integrable Motion of Two Interacting Curves and Two-layer Generalized Heisenberg Ferromagnet Equations

In the present paper, we study the integrable 2-layer generalized Heisenberg ferromagnet equation (HFE). The relation between this generalized HFE and differential geometry of curves is established. Using this relation we found the geometrical equivalent counterpart of the 2-layer spin system which is the 2-component KdV equation. Finally, the gauge equivalence between these equations is established.

nlin.SI↗

arXiv2016

Integrable motion of two interacting curves, spin systems and the Manakov system

Integrable spin systems are an important subclass of integrable (soliton) nonlinear equations. They play important role in physics and mathematics. At present, many integrable spin systems were found and studied. They are related with the motion of 3-dimensional curves. In this paper, we consider a model of two moving interacting curves. Next, we find its integrable reduction related with some integrable coupled spin system. Then we show that this integrable coupled spin system is equivalent to the famous Manakov system.

nlin.SI↗

arXiv2016

Integrable geometric flows of interacting curves/surfaces, multilayer spin systems and the vector nonlinear Schrödinger equation

In this paper, we study integrable multilayer spin systems, namely, the multilayer M-LIII equation. We investigate their relation with the geometric flows of interacting curves and surfaces in some space $R^{n}$. Then we present their the Lakshmanan equivalent counterparts. We show that these equivalent counterparts are, in fact, the vector nonlinear Schrödinger equation (NLSE). It is well-known that the vector NLSE is equivalent to the $Γ$-spin system. Also, we have presented the transformations which give the relation between solutions of the $Γ$-spin system and the multilayer M-LIII equation. It is interesting to note that the integrable multilayer M-LIII equation contains constant magnetic field ${\bf H}$. It seems that this constant magnetic vector plays an important role in theory of "integrable multilayer spin system" and in nonlinear dynamics of magnetic systems. Finally, we present some classes of integrable models of interacting vortices.

nlin.SI↗

arXiv2016

Darboux transformations and exact soliton solutions of integrable coupled spin systems related with the Manakov system

We construct a Darboux transformation of a general $su(3)$-valued spin system called the $Γ$-spin system. Using this Darboux transformation we derive a recursive formula for the soliton solutions of this spin system. Then using these results we present explicit expressions for the 1-soliton solution of the coupled $su(2)$-valued spin systems, namely, of the coupled M-LIII equation.

nlin.SI↗