This paper concerns the Cauchy problem for the Kadomtsev-Petviashvili equation \[ u_t + u_{xxx}+\sigma \partial^{-1}_xu_{yy}=\partial_xu^2, \quad (x,y)\in R^2, \;t\in R, \]
\[ u(0,x,y)=u_0(x,y) , \quad (x,y)\in R^2, \] where \(\sigma=\pm 1\) and \(\partial^{-1}_x=\int^x_{-\infty}dx'\). It is known that the above equation is the KP-I equation and KP-II equation for \(\sigma=-1\) and \(\sigma=1\) respectively. The main result of the paper is that if \(\partial^{-1}_x u_0\in H^{7}\cap H^{5,4}\) and the norm \(\varepsilon:=\|\partial^{-1}_x u_0\|_{H^{5,4}}\) is sufficiently small, then the Cauchy problem above has a unique global solution \(u\in C([0,\infty);H^{6})\) such that \(\|\partial_x u\|_{L^{\infty}}\leq C\varepsilon(1+t)^{-1}\), and there exist \(V\in L^{\infty}\) and \(\gamma\in (0, \frac{1}{100})\) such that \(\|\mathcal{F}\mathcal{U}(-t)u_x(t)-V\|_{L^{\infty}}\leq C\varepsilon^2 t^{-\frac{\gamma}{2}}\) for \(t\geq 1\).