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

$$\begin{aligned} u_{t}-\beta \partial _{x}^{3}u-\gamma \partial _{x}^{-1}u+\frac{1}{k+1}\left( u^{k+1}\right) _{x}=0,k\ge 5 \end{aligned}$$

with $\beta \gamma <0,\gamma >0$. Firstly, we prove that the Cauchy problem for the generalized Ostrovsky equation is locally well-posed in $H^{s}(\mathbf{R})\left( s>\frac{1}{2}-\frac{2}{k}\right)$. Then, we prove that the Cauchy problem for the generalized Ostrovsky equation is locally well-posed in $X_{s}(\mathbf{R}): =\Vert f\Vert _{H^{s}}+\left\| {\mathscr {F}}_{x}^{-1}\left( \frac{{\mathscr {F}}_{x} f(\xi )}{\xi }\right) \right\| _{H^{s}}\left( s>\frac{1}{2}-\frac{2}{k}\right) .$ 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 $X_{s}(\mathbf{R})(s>\frac{3}{2})$. The main difficulty is that the phase function of Ostrovsky equation with negative dispersive $\beta \xi ^{3}+\frac{\gamma }{\xi }$ possesses the zero singular point.