Long-time asymptotic behavior for the nonlocal nonlinear Schr?dinger equation in the solitonic region

In this paper, we extend the ■-steepest descent method to study the Cauchy problem for the nonlocal nonlinear Schr?dinger(NNLS) equation with weighted Sobolev initial data■ ,where ■. Based on the spectral analysis of the Lax pair, the solution of the Cauchy problem is expressed in terms of the solution of a Riemann-Hilbert problem, which is transformed into a solvable model after a series of deformations. We further obtain the asymptotic expansion of the solution to the Cauchy problem for the NNLS equation in the solitonic region. The leading term is soliton solutions, the second term is the interaction between solitons and dispersion, and the error term comes from a corresponding ■-problem.Compared with the asymptotic results on the classical NLS equation, the major difference is the second and third terms of the asymptotic expansion for the NNLS equation, which were affected by a function depending on the scattering data and the stationary phase point.