Power Expansions for the Self-Similar Solutions of the Modified Sawada–Kotera Equation

The fourth-order ordinary differential equation that denes the self-similar solutions of the Kaup–Kupershmidt and Sawada–Kotera equations is studied. This equation belongs to the class of fourth-order analogues of the Painlevé equations. All the power and non-power asymptotic forms and expansions near points $z = 0$, $z = \infty$ and near an arbitrary point $z = z_0$ are found by means of power geometry methods. The exponential additions to the solutions of the studied equation are also determined.