  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
$(1+x^2)^{-1/2-0}H^1(\mathbb{R}^2)$ and perturbations in $H^1(\mathbb{R}^2)\cap
\partial_xL^2(\mathbb{R}^2)$.
