Long-time asymptotic behavior for the nonlocal nonlinear Schrödinger equation in the solitonic region

In this paper, we extend partial derivative<overline>steepest descent method to study the Cauchy problem for the nonlocal nonlinear Schr & ouml;dinger (NNLS) equation with weighted Sobolev initial data %and finite density initial data iq(t )+ q(xx )+ 2 sigma q(2)(x,t)q<overline>(-x,t)=0, q(x,0)=q(0)(x), where q(0) is an element of H- (1,1)(R). Based on the spectral analysis of the Lax pair, the solution of the Cauchy problem is expressed in terms of solutions of a Riemann-Hilbert problem, which is transformed into a solvable model after a series of deformations. Finally, we obtain the asymptotic expansion of the Cauchy problem for the NNLS equation in solitonic region. The leading order term is soliton solutions, the second term is the error term is the interaction between solitons and dispersion, the error term comes from the corresponding partial derivative<overline> equation. Compared to the asymptotic results on the classical NLS equation, the major difference is the second and third terms in asymptotic expansion for the NNLS equation were affected by a function for the stationary phase point .