The nonlinear Schrödinger equation is a well-known partial differential equation, which provides a successful model in nonlinear optic theory, as well as other applications. In this dissertation, following a survey of mathematical literature, the geometric theory of differential equations is applied to the nonlinear Schrödinger equation.
The main result of this dissertation is that the known list of conservation laws for the nonlinear Schrödinger equation is complete. Two theorems are proven, together showing that there is only one equivalence class of conservation law characteristics of the nonlinear Schrödinger equation for each order greater than or equal to three. The known list as given by Faddeev and Takhtajan consists of a single conservation law for each order and a theorem of Olver provides a one-to-one correspondence between equivalence classes of conservation laws and equivalence classes of their characteristics. It follows that the known list of conservation laws of the nonlinear Schrödinger equation is complete.