The Cauchy problem for the generalized Ostrovsky equation with negative dispersion

Abstract

This paper is devoted to studying the Cauchy problem for the generalized Ostrovsky equation \begin{eqnarray*} u_{t}-β\partial_{x}^{3}u-γ\partial_{x}^{-1}u+\frac{1}{k+1}(u^{k+1})_{x}=0,k\geq5 \end{eqnarray*} with $\beta \gamma <0,\gamma >0$. Firstly, we prove that the Cauchy problem for the generalized Ostrovsky equation is locally well-posed in $H^s(\mathbb{R})\left(s>\frac{1}{2}-\frac{2}{k}\right)$. Then, we prove that the Cauchy problem for the generalized Ostrovsky equation is locally well-posed in $X_s(\mathbb{R}):=\Vert f{\Vert }_{H^s}+{\Vert {\mathcal{F}}_x^{-1}\left(\frac{{\mathcal{F}}_{x}f(\xi )}{\xi }\right)\Vert }_{H^s}\left(s>\frac{1}{2}-\frac{2}{k}\right).$ Finally, we show that the solution to the Cauchy problem for generalized Ostrovsky equation converges to the solution to the generalized KdV equation as the rotation parameter $\gamma$ tends to zero for data belonging to $X_s(\mathbb{R})(s>\frac{3}{2})$. The main difficulty is that the phase function of Ostrosvky equation with negative dispersive $\beta {\xi }^3+\frac{\gamma }{\xi }$ possesses the zero singular point.