Soliton resolution for the Harry Dym equation with weighted Sobolev initial data

The soliton resolution for the Harry Dym equation is established for initial conditions in weighted Sobolev space $H^{1,1}(mathbb{R})$. Combining the nonlinear steepest descent method and $bar{partial}$-derivatives condition, we obtain that when $frac{y}{t}<-epsilon(epsilon>0)$ the long time asymptotic expansion of the solution $q(x,t)$ in any fixed cone begin{equation} Cleft(y_{1}, y_{2}, v_{1}, v_{2}right)=left{(y, t) in R^{2} mid y=y_{0}+v t, y_{0} inleft[y_{1}, y_{2}right], v inleft[v_{1}, v_{2}right]right} end{equation} up to an residual error of order $mathcal{O}(t^{-1})$. The expansion shows the long time asymptotic behavior can be described as an $N(I)$-soliton on discrete spectrum whose parameters are modulated by a sum of localized soliton-soliton interactions as one moves through the cone and the second term coming from soliton-radiation interactionson on continuous spectrum.