SOME COMMENTS ON THE RATIONAL SOLUTIONS OF THE ZAKHAROV-SCHABAT EQUATIONS

We describe a family of the rational solutions of the Zakharov-Schabat equations. This family is characterized by extremely simple superposition principle, following directly from the Darboux-invariance of the Zakharov-Schabat equations proved in the works [1, 4]. Particularly we present an infinite sequence of polynomials P n (x, y, t, t 4, ..., t m), m≤n, so that the formula {Mathematical expression} where c i are the arbitrary constants, represents some class of solutions of the Kadomtcev-Petviashvily equation. The paramters t 4, ..., t K represent the explicit action of the commuting flows, related with the Zakharov-Schabat operators of the higher order, on the solutions of the K-P equation, and can be fixed independently in each P i. The polynomials P n are closely related with the second Waring formular well known in algebra. This relation imposes some specific constraints on the motion of the N particle Moser-Calogero system generated by P n. © 1979 D. Reidel Publishing Company.