Theoretical Study Of Formation,Propagation And Stability Of Solitons In Nonlinear Media

Solitons are formed by the balance of the dispersion or diffraction of the optical wave or the matter wave and the nonlinear effect of the media.Their waveshapes,amplitudes and other properties do not change with the time or space.It is demonstrated that the interactions of dark spatial optical solitons can be used into the creation of various kinds of all-optical switches.Solitons have attracted the attention of many physicists in recent years because of their potential applications.In this paper,the existence and stability of solitons under the models of nonlinear optics and Bose-Einstein condensate are studied,and their properties of dynamic propagation are also investigated.In this paper,the solitons in four models of nonlinear Schr?dinger equation are studied,namely the model with bandgap structure in uniform self-defocusing media,the model which overcomes the ctitical collapse in two-dimensional self-foucsing nonlinear media,the model of non-uniform self-defocusing nonlinear media,and the model of nonlinear media under fractional diffraction.Firstly,we theoretically solve the exact solution or approximate solution of the model,and then use numerical program to calculate the numerical solutions of the solitons.We derived the formula of linear stability analysis theoretically,and verified the stability of the obtained numerical solutions by numerical program according to this method,and obtained their stability region.Finally,the finite difference time domain method was used to study their dynamic properties.The original results in this paper are as follows:Firstly,in the model with bandgap structure in uniform self-defocusing nonlinear media,we mainly study the bandgap dark localized modes,including the dark gap solitons and dark gap soliton clusters in both one-dimensional and two-dimensional space.The dark gap solitons in two-dimensional space have not been studied before this paper,and the research on the dark gap soliton cluster is also blank.We study the related linear Bloch band structures,the waveshapes and amplitudes of dark solitons and dark soliton clusters in it.It is found that the chemical potential has a great influence on the waveforms and amplitudes of these dark gap solitons and dark gap soliton clusters.We use the method of linear stability analysis to find the stable regions of the above dark solitons and dark soliton soliton clusters,and observe the properties of their dynamic evolution.Secondly,we overcome the critical collapse of the model in two-dimensional homogeneous self-focusing nonlinear media,and verify that such model can supports stable gap soliton structures in fractional diffraction media,including single soliton,soliton clusters and vortex soliton clusters.The critical collapse in the two-dimensional space is an important scientific problem,and the solution of this problem is very important for the formation of stable high-dimensional solitons.Such critical collapse is overcome in fractional diffraction media by using cubic-quintic competing nonlinearities and optical lattices.It is found that the diffraction order can obviously affect the waveform and stability of solitons.In addition,the position of the soliton structure in the band gap would also affect the waveform and stability of the soliton obviously.Thirdly,in the model of non-uniform self-defocusing nonlinear media,we mainly study the flat-top solitons,the transition from gaussian solitons to flat-top solitons,the modulated solitons and the soliton clusters.It is found that the width of the flat-top soliton is mainly determined by the width of the bottom of the nonlinearity.The influence of chemical potential on the width of the flat-top region is also obvious.For the flat-top solitons in this model,we also study the stability of the fundamental modes and excited modes in both one and two dimensions.In addition,we study the transition region of one-dimensional fundamental solitons and two-dimensional vortex solitons in the parameter space.In the study of modulated solitons and soliton clusters,we find that the positions of the minimum points of nonlinearities are exactly the positions of the humps of the modulated solitons and soliton clusters.If the nonlinear coefficient is set reasonably and the computational space is large enough,the number of the humps of the modulated solitons can become large.We also study the stability of these modulated solitons and soliton clusters.At last,in the nonlinear model under fractional diffraction,we study the solitons in nonlinear lattice,solitons in high-order nonlinearity,and linearly coupled solitons.In the study of solitons in nonlinear lattice,we found that the propagation constant and Lévy index can significantly affect the waveform,amplitude and stability of the solitons in nonlinear lattice.In the study of the model with high-order nonlinearity,we find that the smaller the Lévy index is,the more obvious the sub-peak modulation of the gap soliton with high-order nonlinearity is.Furthermore,the position and order of the band gap have significant effects on the sub-peak modulation of solitons.In addition,we also study the stability of the bandgap soliton in the higher order nonlinearity.In the study of the linear coupled model,we mainly study the asymmetric soliton pairs.It is found that the linear coupling coefficient,Lévy index and chemical potential have significant effects on the structure and stability of these coupled solitons.