We consider the Cauchy problem for the rotation-modified Kadomtsev-Petviashvili (RMKP) equation
$ \begin{align*} \partial_{x}\left(u_{t}-\beta\partial_{x}^{3}u +\partial_{x}(u^{2})\right)+\partial_{y}^{2}u-\gamma u = 0 \end{align*} $
in the anisotropic Sobolev spaces $ H^{s_{1},s_{2}}(\mathbb{R}^{2}) $. When $ \beta <0 $ and $ \gamma >0, $ we prove that the Cauchy problem is locally well-posed in $ H^{s_{1}, s_{2}}(\mathbb{R}^{2}) $ with $ s_{1}>-\frac{1}{2} $ and $ s_{2}\geq 0 $. Our result considerably improves the Theorem 1.4 of R. M. Chen, Y. Liu, P. Z. Zhang(Transactions of the American Mathematical Society, 364(2012), 3395–3425.). The key idea is that we divide the frequency space into regular region and singular region. We further prove that the Cauchy problem for RMKP equation is ill-posed in $ H^{s_{1},0}(\mathbb{R}^{2}) $ with $ s_{1}<-\frac{1}{2} $ in the sense that the flow map associated to the rotation-modified Kadomtsev-Petviashvili is not $ C^{3} $. When $ \beta <0,\gamma >0, $ by using the $ U^{p} $ and $ V^{p} $ spaces, we prove that the Cauchy problem is locally well-posed in $ H^{-\frac{1}{2},0}(\mathbb{R}^{2}) $.
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