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

TL;DR

This paper establishes the long-time asymptotic behavior of solutions to the Harry Dym equation with weighted Sobolev initial data, showing they decompose into solitons and radiation with precise error estimates.

Contribution

It introduces a method combining nonlinear steepest descent and $ar{ ext{D}}$-derivatives to analyze the soliton resolution for the Harry Dym equation with weighted Sobolev initial conditions.

Findings

Asymptotic expansion of solutions in fixed cones with $ ext{O}(t^{-1})$ error

Decomposition into $N(I)$-solitons modulated by soliton interactions

Characterization of soliton-radiation interactions on continuous spectrum

Abstract

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} C\left(y_{1}, y_{2}, v_{1}, v_{2}\right)=\left\{(y, t) \in R^{2} \mid y=y_{0}+v t, y_{0} \in\left[y_{1}, y_{2}\right], v \in\left[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.