On the integrability of nonlinear partial differential equations

We investigate the integrability of Nonlinear Partial Differential Equation s (NPDEs). The concepts are developed by first discussing the integrability of the KdV equation. We proceed by generalizing the ideas introduced for t he KdV equation to other NPDEs. The method is based upon a linearization pr inciple that can be applied on nonlinearities that have a polynomial form. The method is further illustrated by finding solutions of the nonlinear Sch rodinger equation and the vector nonlinear Schrodinger equation, which play an important role in optical fiber communication. Finally, it is shown tha t the method can also be generalized to higher dimensions. (C) 1999 America n Institute of Physics. [S0022-2488(99)01904-0].