Splitting phenomenon of a higher-order shallow water theory associated with a longitudinal planetary waves

•A higher order approximation for the Cauchy-Poisson free boundary problem associated with a nonstationary motion of a perfect incompressible fluid circulating around the equatorial plane of a planet is considered.•It is shown that the above free boundary problem admits two functionally independent systems, while the classical problem for the flat bottom admits only one system.•Additionally, the singular exact solutions are provided for the exact mathematical model describing the above longitudinal planetary waves.

The Cauchy-Poisson free boundary problem associated with a nonstationary motion of a perfect incompressible fluid circulating around the equatorial plane of a planet is considered. It is shown that the corresponding theory of a higher-order shallow approximation admits two functionally independent systems, while the classical problem for the flat bottom admits only one system.