Integrable Management Of Soliton In Quasi-one Dimensional Bose-Einstein Condensates

As the first realization of Bose-Einstein Condensates(BEC) in experiment,it becomes one of the research center topics.In the Bose-Einstein Condensates of ultra-cold atoms,all the atoms occupy the same quantum state which can be described by a macroscopic wave function.At the M-F theory level the macroscopic wave function satisfies the Nonlinear Schrodinger equation(NLSE) or called Gross-Pitaevskii equation (GPE) which supports soliton solutions.The dynamics of matter-wave Solitons have been extensively studied both in experiment and theory in recent years.However,at the most cases,it is difficult for us to get the exact solutions of the non-autonomous NLSE which is determined by BEC system in the presence of the time-dependent magnetic fields and external potentials.In this paper,we reduce the non-autonomous NLSE to a standard one by the usage of the integrability condition.From the solutions to the standard NLSE,we can easily get the exact solutions of non-autonomous NLSE.In this work,we investigate the dynamics of solitons in BEC in the presence of the magnetic field and the external confined potential whose relationship is confined by the integrability condition.In chapter two,we discuss the scheme of integrable management of soliton in BEC by the Feshbach resonance.Firstly,we investigate the integrable management on dark soliton in BEC.Our analysis shows that the dynamics of soliton is determined by the variation of scatter length between atoms(the variation of nonlinearity).When the nonlinearity is enhanced the amplitude of dark soliton in repulsive BEC is increasing by the integrable management,and verse visa.Sencondly,we discuss the integrable management on bright soliton in attractive BEC.We find the solitons can be compressed with very high local density of atoms by the integrable management.And its dynamics is determined by the specific form of the time-dependent magnetic field.In addition,we introduce the colliding soliton solution of the standard NLSE which describes the colliding process of solitons to study the interference between solitons during the colliding process.And the result shows that the interference during the colliding of solitons is not changed by the integrable management of the magnetic field and the confined harmonic potential.When the distance between solitons become larger in the processing,the behavior of each soliton is similar to a single soliton by the integrable management.In chapter three,we discuss the integrable management of soliton in the presence of dissipation,and analyze the dynamics of the soliton by compare with the case of no integrable management.The width of the dark soliton is modulated periodically and it's amplitude is enhanced by the sinusoidal magnetic field and corresponding trapping potential when the dissipation coefficient is positive(or gain).When the dissipation coefficient is negative,the amplitude of the dark soliton is enhanced at the beginning. However,the soliton will decay eventually due to the dissipation.The amplitude and the width of bright soliton can be modulated periodically during the process.As the case of no dissipation case,the amplitude can be compressed with very high local density.At the present of loss,the amplitude of the bright soliton can be compressed at the beginning. And the amplitude of bright soliton will decreasing to one and the width is expanding due to the dissipation eventually.