Study On The Mixed Soliton Solutions Of The-Symmetric Nonlocal Nonlinear Schr?Dinger Equation

Since the rise of parity-time()symmetric quantum theory,the notion of symmetry has been extended to such fields as optics,electricity and material science,and it has also received attention in the field of integrable systems and soliton theory.For the-symmetric nonlocal nonlinear Schr?dinger(NNLS)equation,in this thesiswe use the Darboux transformation method to obtain two new types of the exponential-rational mixed soliton solutions,and study the soliton interaction properties.Via the asymptotic analysis,we find that the first type of solution contains two exponential solitons and two rational solitons,and it can describe the standard elastic interactions of these four solitons.For the second type of solution,we improve the asymptotic analysis technique and obtain four mixed asymptotic solitons with curved center trajectories.They have the t-dependent velocities and the phase shift upon an interaction also changes with the time in the logarithmic manner.However,the interactions among the mixed solitons are still elastic in the sense that the amplitudes of all asymptotic solitons remain invariant before and after interactions and their velocities have always the same values at t and-t.Considering each soliton can display the dark or anti-dark profile,we classify the soliton interactions for two types of mixed soliton solutions,and obtain the corresponding parametric conditions.This study reveals more soliton solutions and the relevant properties which are different from those in the local nonlinear Schr?dinger equation,and the improved asymptotic analysis method can also apply to study the asymptotic behavior of soliton solutions of other nonlocal integrable equations.