Power and exponential expansions of solutions for the equation of the $K_{II}$ hierarchy

M. S. Nialepka

Trudy Instituta matematiki, Tome 16 (2008) no. 2, pp. 97-104

Résumé

Consider fourth order ordinary differential equation, which is a particular case of the $K_{II}$ hierarchy when $n=1$. We applied the Newton polygons method, evolved by A. D. Bruno. We used that method to find power, exponential expansions and their exponential additions for solutions of the equation. As the result we received a four-parametric family of holomorphic expansions, four asymptotic expansions, four families of polar expansions, holomorphic expansions around infinity and their exponential additions. When $\beta=0$ we had the particular cases of the expansions listed above, noted that for the holomorphic near infinity expansions the particular case were four exponential expansions.