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 $$u = 2\partial _x^2 Log\left( {\sum\limits_{i = 1}^N {c_i P_i } } \right)$$ 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.