Application Of Darboux Transform And Bilinear Method To Nonlocal Nonlinear Equations

Among many natural phenomena the nonlocality is a common phenomenon.Some theoretical analysis and numerical simulation show that the nonlocality can eliminate the collapse of the wave and greatly improve the interaction between the dark solitons.Currently,many solutions of local nonlinear equations have been obtained by using Darboux transformation(DT)and bilinear method,but there are few of work to solve nonlocal equations by DT and bilinear method.This paper will do the following research on this issue.In the second chapter,we mainly solve the nonlocal nonlinear Schr?dinger equation(NNLSE)with the self-induced PT-symmetric potential by DT.The N-fold DT of NNLSE is derived with the helping of the gauge transformation between the Lax pairs.Finally,some novel exact solutions are obtained including the bright soliton,breather wave soliton.In particularly,the dynamic features of one-soliton,two-soliton,three-soliton solutions and the elastic interactions between the two solitons are displayed in this chapter.So far,the generalized nonlocal nonlinear Hirota(GNNH)equation has been widely concerned,it can be regarded as the generalization of the nonlocal Schr?dinger equation,and can be reduced to a nonlocal Hirota equation.Therefore,we also study a GNNH equation and its determinant representation of N-order Darboux transformation in this chapter.Then some novel exact solutions including the breather wave solitons,bright solitions,some characteristics of solitary wave and interactions are considered.In particularly,the dynamic features of one-soliton,two-soliton solutions and the elastic interactions between the two solitons are displayed.We find that unlike the local case,the q(x,t)andq~*(-x,t)of the GNNH equation have some novel characteristics of solitary wave,which are different from the classical Hirota equation.In the third chapter.Firstly,we investigate the bilinear form of the nonlocal complex integrable modified Kortewegde Vries(mKdV)equation,the one and two soliton solutions of the nonlocal complex integrable mKdV equation are obtained by using the bilinear method.Then we select the appropriate parameters and draw the exact solution of the equation by using Maple software.Secondly,we also study the bilinear form of the nonlocal integrable Schr?dinger equation with variable coefficients,and obtain its one soliton solution and two soliton solution.Specifically,we succeed to bilinearize the generalized nonlocal Gross-Pitaevskii(NGP)equation with an arbitrary time-dependent linear potential through a nonstandard procedure and present more general bright soliton solutions,which describes the dynamics of soliton solutions in quasi-one-dimensional Bose-Einstein condensations.Under some reasonable assumptions,one-bright-soliton and two-bright-soliton solutions are constructed analytically by the improved Hirota method.From the gauge equivalence,we can see the difference between the solutions of the nonlocal variable coefficient Schr?dinger equation and the solutions of the local variable coefficient Schr?dinger equation.