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| We analyze the dynamical behavior of the multisoliton train in adiabatic approximation of the perturbed Manakov system (MS)
\begin{equation}
i\textbf{u}_t+ \frac{1}{2}\textbf{u}_{xx}+ (\overline{\textbf{u}}, \textbf{u}) \textbf{u}=V(x)\textbf{u}(x,t)
\end{equation}
with composite external potentials of kind $V(x)=\sum_{s=1}^N c_s {\rm sech}^2 (x-x_s)$
where $x_s$ locate the positions of the small-amplitude ($|c_s|\ll 1$) wells/humps.
We analyze the dynamics of multisoliton trains of Cauchy problem composed by MS and the initial condition \[ \textbf{u}(x,t=0) = \sum_{k=1}^{N} u_{k; \rm 1s}(x,t=0) \textbf{n}_k,\]
where $u_{k; \rm 1s}(x,t)$ is the 1-soliton solution of the scalar nonlinear Schrodinger equation with given velocity, amplitude, phase, and position, and $\textbf{n}_k$ is normalized polarization vector (for details see [1]).
We show that the dynamics of the multisoliton train is modeled by a perturbed complex Toda chain for the train parameters which generalize the results of [1]. Combining the analytic and numerical approach we focus also on the perturbation effects on the asymptotically free behavior, the bound state regime as well as on the mixed asymptotic regimes of the soliton trains and the transitions between them under the external potential. The results obtained extend the ones in Refs. [2,3,4].
The investigation is partially supported by the National Scientific Foundation of the Bulgarian Ministry of Education, Youth, and Science under grant DDVU02/71.
$\textbf{References}$
[1] V.S. Gerdjikov, E.V. Doktorov, and N.P. Matsuka. N-soliton Train and Generalized Complex Toda Chain for Manakov System, Theor. Math. Phys. 151(3), 762--773 (2007).
[2] M.D. Todorov and C.I. Christov. Impact of the Large Cross-modulation Parameter on the Collision Dynamics of Quasi-particles Governed by Vector NLSE, Math. Comp. Simul. 80(1), 46--55 (2009).
[3] V.S. Gerdjikov and M.D. Todorov. "On the Effects of Sech-like Potentials on Manakov Solitons,'' in 5th AMiTaNS'13, AIP CP1561, pp. 75--83, DOI: 10.1063/1.4827216, Melville, NY, 2013.
[4] V.S. Gerdjikov, and M.D. Todorov. "N-soliton Interactions for the Manakov System: Effects of External Potentials,'' in R. Carretero-Gonzalez et al. (eds.), Localized Excitations in Nonlinear Complex Systems, Nonlinear Systems and Complexity 7, pp.147--169, DOI: 10.1007/978-3-319-02057-0$\_$7, Springer International Publishing Switzerland, 2014. |
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