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Two Interacting Surfaces and Curves Corresponding to Periodic Solutions of Manakov System
Ratbay Myrzakulov, Akbota Myrzakul

Last modified: 2017-04-13


In this talk, we want to consider the following Manakov system \cite{key-1}\[iq_{1t}+q_{1xx}+2(|q_{1}|^{2}+|q_{2}|^{2})q_{1}=0, \eqno (1a)\]\[iq_{2t}+q_{2xx}+2(|q_{1}|^{2}+|q_{2}|^{2})q_{2}=0, \eqno (1b)\]where $q_{j}$ are complex functions. The gauge and geometrical equivalent counterpart of the Manakov system (1) is given by \cite{key-2}\[iA_{t}+\frac{1}{2}[A,A_{xx}]+iu_{1}A_{x}+v_{1}[\sigma_{3},A]=0, \eqno(2a) \]\[iB_{t}+\frac{1}{2}[B,B_{xx}]+iu_{2}B_{x}+v_{2}[\sigma_{3},B]=0,\eqno(2b)\]where $u_{j}$ and $v_{j}$  are some real functions (potentials) and\[A=\begin{pmatrix} A_{3}&A^{-}\\ A^{+}&-A_{3}\end{pmatrix}, \quad B=\begin{pmatrix} B_{3}&B^{-}\\ B^{+}&-B_{3}\end{pmatrix},\quad A^{2}=B^{2}=I.\eqno(3)\] It is the 2-layer M-LIII equation. It is well known that the 2-layer M-LIII equation and the Manakov system are integrable by IST.  The Darboux transformation (DT) for the simple periodic "seed"  solution of the Manakov system (1) is presented. Using this DT, the exact solutions of the Manakov system is considered. Next, using the Sym-Tafel formula, the two interacting  surfaces and curves  related with the solutions of the Manakov system were constructed. 
\begin{thebibliography}{1}\bibitem[1]{key-1}S.V. Manakov. On the theory of two-dimensional stationary self-focusing of electromagnetic waves.  Sov. Phys. JETP, {\bf 38}, 248 (1974).\bibitem[2]{key-2}Akbota Myrzakul, Ratbay Myrzakulov. Integrable motion of two interacting curves, spin systems and the Manakov system. International Journal of Geometric Methods in Modern Physics (accepted), (2017). [arXiv:1606.06598]\end{thebibliography}

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