We study the gap solitons and nonlinear Bloch waves of interacting bosons in one-dimensional optical lattices, taking into account the interaction from the weak to the strong limits. It is shown that composition relation between the gap solitons and nonlinear Bloch waves exists for the whole span of the interaction strength. The linear stability analysis indicates that the gap solitons are stable when their energies are near the bottom of the linear Bloch band gap. By increasing the interaction strength, the stable gap solitons can turn into unstable. It is argued that the stable gap solitons can easily be formed in a weakly interacting system with energies near the bottoms of the lower-level linear Bloch band gaps.
In terms of Darboux transformation we investigate the dynamic process of spin wave passing through a magnetic soliton. It causes nonlinear excitations, such as Akhmediev breathers solution and Kuznetsov-Ma soliton. The former case demonstrates a spatial periodic process of a magnetic soliton forming the petal with four pieces. The spatial separation of adjacent magnetic petals increases rapidly, while one valley splits into two and the amplitude of valley increases gradually with the increasing amplitude of spin wave. The other case shows a localized process of the spin-wave background. In the limit case, we get rogue waves and clarify its formation mechanism.
By means of the modified Darboux transformation we obtain some types of rogue waves in two-coupled nonlinear Schrödinger equations. Our results show that the two components admits the symmetry and asymmetry rogue wave solutions, which arises from the joint action of self-phase, cross-phase modulation, and coherent coupling term. We also obtain the analytical transformation from the initial seed solution to unique rogue waves with the bountiful pair structure. In a special case, the asymmetry rogue wave can own the spatial and temporal symmetry gradually, which is controlled by one parameter. It is worth pointing out that the rogue wave of two components can share the temporal inversion symmetry.
We study two types of bright solitons in an attractive Bose-Einstein condensate with a spin-orbit interaction. By solving the coupled nonlinear Schrödinger equations with the variational method and the imaginary time evolution method, fundamental properties of solitons are carefully investigated in different parameter regimes. It is shown that the detuning between the Raman beam and energy states of the atoms dominates the ground state type and spin polarization strength. The soliton dynamics is also studied for various moving velocities for zero and nonzero detuning cases. We find that the shape of individual component solitons can be maintained when the moving speed of solitons is low and the detuning is small in the coupled harmonically trapped pseudo-spin polarization Bose-Einstein condensate.
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