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In this paper, we apply ∂ steepest descent method to study the long time asymptotic behavior for the Cauchy of the three-wave resonant interaction equationwhere n ij are constants. It is shown that the solution of the Cauchy problem can be characterized via a the solution of Riemann-Hilbert problem.In any fixed space-time conewe further compute the long time asymptotic expansion of the solution u(x, t), which implies soliton resolution conjecture and can be characterized with an N (I)-soliton whose parameters are modulated by a sum of localized soliton-soliton interactions as one moves through the region,

In this paper, we consider the Cauchy problem for the modified NLS equationwhere H 2,2 (R) is a weighted Sobolev space. Using nonlinear steepest descent method and combining the ∂-analysis, we show that inside any fixed conethe long time asymptotic behavior of the solution u(x, t) for the modified NLS equation can be characterized with an N (I)-soliton on discrete spectrum and leading order term O(t −1/2 ) on continuous spectrum up to an residual error order O(t −3/4 ).