Soliton Solutions And Stability Analysis In Combined Time-Dependent Magnetic-Optical Potentials

Optical lattices, generated by coherent laser beams illuminating the conden-sate from opposite directions, have been proven as a very efficient tool for con-trolled manipulation of stable, spatially localized matter waves in Bose-Einstein condensates. The experiments have demonstrated that one-dimensional optical lattices with spatially modulated nonlinearity can support stable matter wave multisoliton solutions in terms of Mathieu and elliptic functions. This periodic lattices also especially efficient in supporting matter-wave solitons and have the modulation amplitude which can take the periodic or switching functionsA unique property of optical lattices is that periodic lattices are combined with a linear trapping potential, which can be continuously tuned by linearly increasing of the scattering length in time in an external magnetic field near the Feshbach resonance (FR). Ones shall refer to this potential as the combined linear-lattice potential. Nonautonomous soliton and localized wave tapped into combined linear-lattice potential can maintain their overall shapes but allow their widths, amplitudes and the pulse center to change according to management of the system’s parameters. It is very necessary that research solitons trap into the combined linear-lattice potential. Another unique property of optical lattices, as realized in experiments, is that the periodic lattice is accompanied by a harmonic confining potential. Ones shall refer to this potential as the combined harmonic-lattice potential. The combined harmonic-lattice potential is also interesting for possible applications in quantum information and the dynamics of Bose-Einstein condensate solitons.The optical lattices, linear-lattice potential and harmonic-lattice potential are called combined magnetic-optical potentials, which potentials may be real-ized in Bose-Einstein condensates by tuning the external magnetic field and the optically controlled interactions using the FR technique. A most interesting issue is the construction of exact localized nonlinear wave solutions in systems of the combined time-dependent magnetic-optical potentials. By appropriate selection of the system parameters, one can control the shape, speed, matter-wave soliton propagation paths and so on.In this paper, we investigate the explicit matter-wave soliton solutions of the cubic-quintic nonlinear Schrodinger equation with spatiotemporal modula-tion of the nonlinearities and potentials. We present a new transformation re-lation and construct some localized cubic-quintic nonlinearities and many kinds of potentials, including optical lattice potential and combined time-dependent magnetic-optical potentials in the form of linear-lattice, harmonic-lattice and harmonic-linear-lattice ones. Also, corresponding analytical localized soliton so-lutions in terms of Mathieu and elliptic functions are studied, such as snakelike solitons, moving breathing solitons, oscillating solitons and so on. We perform some numerical simulations to the exact solitons trap in the optical lattice poten-tial and combined time-dependent magnetic-optical potentials by means of the split step Fourier-transform method. And some stable solitons are found. This thesis is organized as follows:(1) Solitons in the optical lattice potential and stability analysis. We study the dynamics of the exact localized nonlinear waves in the optical lattice. The results of the stability analysis show that the soliton with n=1are stable, while the higher solitons (n>1) are unstable;(2) Snakelike solitons in the combined time-dependent linear-lattice potential and stability analysis. We study the dynamics and possible physical applications of the snakelike solitons. And then, we check the stability of the solutions by employing the numerical split-step Fourier method, which has direct simulations, the perturbations in the constraint conditions g(x,t) and G(x,t) and add pertur-bation with an initial white noise of level1%. The results show that the snakelike soliton with n=1are stable, while the higher solitons(n>1) are unstable;(3) Breathing solitons and moving breathing solitons in the combined time-dependent harmonic-lattice potential and stability analysis. Firstly, we analyze the harmonic-lattice potential and breathing solitons and moving breathing soli-tons. And when the elliptic modulus m more and more close to1(m≠1), the shapes of the functionΦ(X) happen some changes, e.g., the sharp peak becomes more and more explanate with n=1. For a small elliptic modulus m, the shapes of the breathing solitons may exhibit some interesting features, i.e., one breath-ing soliton dividing into two. The results of the stability analysis show that the breathing soliton with n=1are stable, while the excited states (n>1) are unstable. Ones find that the moving breathing solitons in the combined time-dependent harmonic-lattice with β4≠0are all unstable;(4) Oscillating solitons in the combined time-dependent harmonic-linear-lattice potential and stability analysis. By choosing the parameters, harmonic-linear-lattice potential is obtained. We also find the explicit oscillating soliton solutions of the cubic-quintic nonlinear Schrodinger equation. The dynamics and possible physical applications of the oscillating solitons are investigated. Finally, the re-sults of the stability analysis of oscillating solitons show that soliton in the ground state is stable and the excited states are unstable.Finally, we summarize the main results and give an outlook of the future study in this field.