N-Soliton Interactions for the Manakov System: Effects of External Potentials
We analyze the dynamical behavior of the N-soliton train in adiabatic approximation of the Manakov system (MS) perturbed by three types of external potentials: periodic, quadratic and quartic ones. We show that the dynamics of the N-soliton train is model
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Abstract We analyze the dynamical behavior of the N -soliton train in adiabatic approximation of the Manakov system (MS) perturbed by three types of external potentials: periodic, quadratic and quartic ones. We show that the dynamics of the N -soliton train is modeled by a perturbed complex Toda chain for certain choices of the train parameters and for small magnitudes of the intensities of the potentials. Possible applications of these results for Bose-Einstein condensates are discussed.
1 Introduction The Gross-Pitaevski (GP) equation and its multicomponent generalizations are important tools for analyzing and studying the dynamics of the Bose-Einstein condensates (BEC) , see the monographs [25, 29, 42] and the numerous references therein among which we mention [4–6, 15, 21, 22, 26, 31, 36, 38, 41]. In the 3dimensional case these equations can be analyzed solely by numerical methods. If we assume that BEC is quasi-one-dimensional then the GP equations mentioned above reduce to to the nonlinear Schrödinger (NLS) equation 1 i ut C uxx C juj2 u.x; t/ D 0; 2
(1)
V.S. Gerdjikov () Institute for Nuclear Research and Nuclear Energy, Bulgarian Academy of Sciences, 72 Tzarigradsko chaussee, Blvd., 1784 Sofia, Bulgaria e-mail: [email protected] M.D. Todorov Department of Applied Mathematics and Computer Science, Technical University of Sofia, 8 Kliment Ohridski, Blvd., 1000 Sofia, Bulgaria e-mail: [email protected] R. Carretero-González et al. (eds.), Localized Excitations in Nonlinear Complex Systems, Nonlinear Systems and Complexity 7, DOI 10.1007/978-3-319-02057-0__7, © Springer International Publishing Switzerland 2014
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and its vector generalizations (VNLS) 1 i ut C uxx C .u ; u/u.x; t/ D 0: 2
(2)
perturbed with external potential. Until now two types of integrable VNLS equations are known. The first and oldest one is known as the Manakov model (MM) [37] – two-component VNLS which is easily generalized also to any number of components. Most of our results below concern this model. The second VNLS was discovered by Kulish and Sklyanin [34]. Its importance for the spin-1 and spin-2 BEC was discovered in [26, 27, 40, 48, 49]. The Lax representation, N -soliton solutions and the fundamental properties of the MM have been well known for long time now [37], see also [1, 19] and the numerous references therein. The Kulish-Sklyanin model [34], though less popular have also been thoroughly investigated. Its Hamiltonian properties, N -soliton solutions and their interactions have also been derived [9, 23, 32, 33]. The analytical approach to the N -soliton interactions was proposed by Zakharov and Shabat [39, 51] for the scalar NLS. They treated the case of the exact N -soliton solution where all solitons had different velocities. They calculated the asymptotics of the N -soliton solution for t ! ˙1 and proved that both asymptotics are sums of N 1-soliton solutions with the same sets of amplitudes and velocities. The effects of the interaction were shifts in the relative cente
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