We study the long time asymptotic behavior for the Cauchy problem of the modified Camassa-Holm (mCH) equation with step-like initial valvewhere A 1 , A 2 are two positive constant. Our main technical tool is the representation of the Cauchy problem with an associated matrix Riemann-Hilbert (RH) problem and the consequent asymptotic analysis of this RH problem.Based on the spectral analysis of the Lax pair associated with the mCH equation and scattering matrix, the solution of the Cauchy problem is characterized via the solution of a RH problem in the new scale (y, t). Further using the Deift-Zhou steepest descent method, we derive different long time asymptotic expansion of the solution u(y, t) in different space-time solitonic regions of ξ = y/t and the different choice of the initial value. We divide the half-plane {(y, t) : −∞ < y < ∞, t > 0} into three asymptotic regions by the different choice of g-function. The corresponding asymptotic approximations can be characterized with the plane wave in genus-0 Region I and II, and hyperelliptic

In this paper, we extend ∂ steepest descent method to study the Cauchy problem for the nonlocal nonlinear Schrödinger (NNLS) equation with weighted Sobolev initial datawhere q 0 (x) ∈ L 1,1 (R) ∩ L 2,1/2 (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 ∂ 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 Imν(ξ) for the stationary phase point ξ.

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