Bifurcation And Qualitative Study Of Travelling Wave Solutions In GNLSE With Saturable Nonlinearity

As social progress and ceaseless scientific research,a large number of nonlinear mathematical models appear in actual engineering and various branches of natural sciences,even,in the domain of social sciences,which waiting for being studied deeply by science workers.The explicit solution expressions of these nonlinear issues are difficult obtained,compare to the case of linear equations.The nonlinear Schrodinger equation(NLSE)is one of the most typical examples,which appears in many applied areas like Fluid Dynamics,Plasma Physics,Protein Chemistry and engineering sciences.It is significant not only in the basic research but also in the engi-neering practicality to research and understand more about the properties of various solutions.In the basic research for the NLSE,it becomes an important topic to study kinds of finity travelling waves which including solitary waves,a lot of techniques and methods have been developed such as the inverse scattering method,Darboux transformation method,Hirota bilinear method,tanh method and so on.The basic idea of these method is that transforming original equations to a resolvable forms,especially, obtaining the peculiar explicit solutions as solitons of the equations un-der the given conditions.Until now,many mathematicians and physicists from inland and overseas have gained large numbers of outcomes.How-ever,except determining the explicit solutions under the given conditions, these methods can't give the integrated explanation about the relationship between parameters and the existence of peculiar finity solutions.Latest researches indicate the theory of bifurcation method and qualitative anal-ysis of dynamical system can make up these deficiencies,even,by using the theoretics of dynamical system,we can get much deeper understanding about explicit solutions of equations.Based mentioned above,by using the bifurcation theory of dynamical system and qualitative analysis of differential equation(in particular,the method of phase analysis for Hamiltonian system),we study the param-eters bifurcation of various travelling wave solutions for following NLSE with saturable nonlinearity: where g(I)is real value non-kerr law nonlinearity function,which reflects saturable nonlinearity character in physics: where the parameter a has the meaning of a ratio of the maximum intensity I_maxto the saturation intensity,I_sat,i.e.,a=I_max/I_sat,and the parameter p＞0 is the saturation index.In allusion to above model,it is proved that there are local solutions for dark solitons and bright solitons when p=1 and p=2 in[26,27],which exhibit explicit solitary wave solutions in the analytical form.However, what about the other more general p? Although those papers point out there still exist above mentioned solitons when p＞2,as we know,there are few furthermore researches for this case.By using the bifurcation theory Of dynamical system and qualitative analysis of differential equation,in particular,the method of phase analysis for Hamiltonian system,under the general saturation index p＞0,we learn adequately about the parameters bifurcation for various trvalling travelling wave of mentioned generalized NLSE,and obtain bifurcation conditions in corresponding four dimensions parameters space(a,p,the soliton velocity v and the intensity q of the background).The result of this paper indicates that saturable nonlinearity models corresponding p＞2 have all kinds of solitary wave solutions and period solutions which include the solutions of models when p≤2,while certain solitons are not exist in these models when p≤2.