Stability of line solitons for the KP-II equation in ${\mathbb{R}}^2$, II

The KP-II equation was derived by Kadmotsev and Petviashvili to explain stability of line solitary waves of shallow water. Recently, Mizumachi (Mem. Amer. Math. Soc. 238 (2015)) has proved nonlinear stability of $1$-line solitons for exponentially localized perturbations. In this paper, we prove stability of $1$-line solitons for perturbations in $(1+x^2{)}^{-1\mathrm{/}2-0}{H}^1({\mathbb{R}}^2)$ and perturbations in $H^1($\mathbb{R}$^2)\cap \partial_xL^2($\mathbb{R}$^2)$.