On the integrability of non-linear partial differential equations

We investigate the integrability of Nonlinear Partial Differential Equations (NPDEs). The concepts are developed by first discussing the integrability of the KdV equation. We proceed by generalizing the ideas introduced for the KdV equation to other NPDEs. The method is based upon a linearization principle that can be applied on nonlinearities that have a polynomial form. The method is further illustrated by finding solutions of the nonlinear Schrödinger equation and the vector nonlinear Schrödinger equation, which play an important role in optical fiber communication. Finally, it is shown that the method can also be generalized to higher dimensions