We present a solution of the Weiss operator family generalized for the case of Rd and formulate a dimensional analogue of the Weiss Theorem. Most importantly, the generalization of the Weiss Theorem allows us to find a subset of null class functions for a partial differential equation with the generalized Weiss operators. | Turk J Math 35 (2011) , 687 – 694. ¨ ITAK ˙ c TUB doi: Multi-dimensional Weiss operators Sergey Borisenok, M. Hakan Erkut, Ya¸sar Polato˘glu, Murat Demirer Abstract We present a solution of the Weiss operator family generalized for the case of Ê d and formulate a d - dimensional analogue of the Weiss Theorem. Most importantly, the generalization of the Weiss Theorem allows us to find a subset of null class functions for a partial differential equation with the generalized Weiss operators. We illustrate the significance of our approach through several examples of both linear and non-linear partial differential equations. Key Words: Partial differential equations, Weiss operators 1. Introduction To investigate the integrability of nonlinear partial differential equations, many different methods have been developed; among them is Painlev´e test [1]. In the framework of this approach, the class of one-dimensional (’scalar’) differential operators was defined first in [2], and in [3] has been applied to solitonic-type PDEs. The family of Weiss operators performs a special kind of ordinary derivative operators of integer order n (n > 0 and for each order only one such an operator exists), and the general solution for each Weiss operator can be found in a very simple form. Following [2], let’s define for a differentiable scalar function φ(x) the class of factorized differential operators as Ln+1 = n d d n n d n V = + j− − V · ··· · + V , dx 2 dx 2 dx 2 (1) j=0 where V = φxx/φx is the so-called pre-Schwarzian. For n = 0 equation (1) produces the ordinary derivative odinger operator L1 = d/dx ; for n = 1 , we get the Schr¨ L2 = d 1 − V dx 2 d 1 + V dx 2 = d dx 2 1 + S; 2 2000 AMS Mathematics Subject Classification: 35A24, 47F05. 687 ˘ ˙ BORISENOK, ERKUT, POLATOGLU, DEMIRER for n = 2 , we get the the Lenard operator L3 = d −V dx d dx d +V dx = d dx 3 + 2S d + Sx , dx where by S we denote the
