FAMILIES OF RATIONAL SOLITON SOLUTIONS OF THE KADOMTSEV-PETVIASHVILI I EQUATION

Families of exact explicit nonsingular rational soliton (lump) solutions of any order to the Kadomtsev-Petviashvili I equation are presented in a compact form. We show that the higher-order lump solutions may exhibit rich patterns on a finite background, but invariably evolve from a vertical distribution at large negative time into a horizontal distribution at large positive time, within an appropriate Galilean transformed frame. A universal polynomial equation is then put forward, whose real roots can accurately determine the lump positions in such a complex multi-lump distribution. We also unveil that there is an intimate relation between certain lump structures and the rogue-wave hierarchy. We expect that this finding may provide a new pathway for understanding the higher-dimensional rogue waves.