Soliton, rational and special solutions of the Korteweg–de Vries hierarchy

Soliton, rational and special solutions for any member of the Korteweg–de Vries hierarchy are presented. It is shown that the soliton solutions are found by means of the Hirota formulae with change of the dispersion relation, special solutions for the Korteweg–de Vries hierarchy are expressed via the transcendents of the first and the second Painlevé hierarchies and rational solutions of members for the Korteveg–de Vries hierarchy can be presented using the Yablonski–Vorobév polynomials.