A simple explicit numerical scheme is proposed for the solution of the Gardner–Ostrovskyequation (ut + cux + α uux + α1u2ux + βuxxx)x = γuwhich is also known as the extended rotation-modified Korteweg–de Vries(KdV) equation. This equation is used for the description of internal oceanic wavesaffected by Earth’ rotation. Particular versions of this equation with zero some ofcoefficients, α, α1, β, orγ are also known in numerous applications. The proposed numericalscheme is a further development of the well-known finite-difference scheme earlier usedfor the solution of the KdV equation. The scheme is of the second order accuracy both ontemporal and spatial variables. The stability analysis of the scheme is presented forinfinitesimal perturbations. The conditions for the calculations with the appropriateaccuracy have been found. Examples of calculations with the periodic boundary conditionsare presented to illustrate the robustness of the proposed scheme.

The well-posedness of the Ostrovsky equation is considered. Local well-posedness for data in $\tilde{H}^s(\mathbb{R})$ $(s\geq-\frac{1}{8})$ and global well-posedness for data in $\tilde{L}^{2}(\mathbb{R})$ are obtained.