| In recent years,the spatial optical solitons has gradually become a research hotspot in the whole word.However,most of the studies have been focused on the solitons in nonlocal nonlinear media.Recently,the studies on spatial optical solitons in competing nonlocal media have gradually increased.However,the study of competing nonlocal solitons is mainly focused on the model in which the refractive index change induced by the beam is equal to the sum of the refractive index change induced by two nonlinear effects respectively.In 2017,JUNG P S et al.found that the refractive index change induced by the molecular orientation effects and thermal effects in nematic liquid crystals is equal to multiply the two refractive index change induced by the two nonlinear effects especially.Since this competing nonlocal model was proposed,a small amount of literature has reported the properties of bright solitons in this model.To date,no literature has reported the properties of dark solitons in this model.In this paper,the model is used to study the stability and beam steering by bright solitons.In addition,the propagation properties of dark solitons are studied by using variational method.The properties of dark soliton in the fractionalcompeting nonlocal nonlinear Schr(?)dinger equation are obtained by using numerical methods.The details of the study are as follows:Firstly,the history of soliton development,soliton types,etc.,the current status of research on solitons in nonlocal nonlinear media,the current situation of research on soliton stability in nonlocal media,and the current situation of research on soliton interactions,as well as the current situation of research on solitons in competing nonlocal nematic liquid crystals are introduced.Secondly,the stability of solitons in competing nonlocal media is studied.Based on the nematic liquid crystal model,the stability of bright solitons in competing nonlocal nematic liquid crystals is analyzed by perturbation method.The equation which describe the perturbation term of the soliton is obtained,and the eigenvalues of the perturbation equation are obtained by using the numerical method.The stability of the solitons is judged from the eigenvalues,and it is found that there are stable optical solitons in the competing nonlocal medium.The eigenvalues of the perturbation terms of ground state solitons,dipole solitons and triode solitons is given.It is found that the nonlocality of molecular orientation,the nonlocality of thermal effect and the thermal nonlinearity coefficient all affect the stability of the solitons.The stability intervals of ground state solitons,dipole solitons and tertiary solitons with different parameters are obtained.Thirdly,the properties of dark solitons in liquid crystals with competing nonlinearities are investigated by the model which is proposed by JUNG P S.Under the condition of the rectangular response function,the relationship between the width of dark soliton and the parameters of the medium is obtained by the variational method.It is found that,when all parameters are fixed and only one of the parameter is changed,the width of the dark solitons first decrease and then increase with the increment of the orientational nonlocality;the solitons width increase monotonously with the increase of the thermal nonlocality;the solitons width decrease monotonously with the increase of the thermal nonlinearity coefficient.It is found that the width of dark solitons is more sensitive to the orientational nonlocality than the other parameters.In addition,the solitons width increase monotonously with the grayness of the solitons.At last,the propagation of the dark solitons is obtained by the split step Fourier algorithm.The analytical results are consistent with the numerical simulations.Fourty,the dark soliton and bright soliton interactions in competing nonlocal media are investigated.The bright soliton interactions in competing nonlocal nematic liquid crystals are numerically simulated,and the soliton interactions are investigated for different beam separation,beam phase,and thermal nonlinearity coefficients.It is found that the initial separation greatly affects the interaction between solitons,which attract each other and form periodic bound states,and when the initial separation is large enough,two soliton will propagate forward in parallel without affecting each other.It is also found that two solitons will exchange energy and even transfer energy from one to another under different thermal nonlinear coefficients.Finally,the dark soliton propagation properties of the fractional competing nonlocal Schr(?)dinger equation are investigated.The soliton solutions of the fractional Schr(?)dinger equation are obtained by using the square operator method,and the variation of soliton width and output power with the material parameters are investigated.It is found that the dark soliton width and power increase with increasing of Levy index,nonlocality of molecular orientation and thermal effct,and decrease with the increase of thermal coefficient.It is also found that the change of the thermal nonlocality has little effect on the width and power of the dark soliton,while the change of the Lévy index,the molecular orientation nonlocaltiy and the value of the thermal nonlinear coefficient greatly affect the initial width and power of the dark soliton. |
