Graduate Program in Mathematics.Gürel, Burak.Yılmaz, Oğuz.2023-10-152023-10-152022MATH 2022 Y56https://digitalarchive.library.bogazici.edu.tr/handle/123456789/19901The thesis is a survey of the I-method. After introducing the method, we discuss the implementation of this method to cubic, defocusing nonlinear Schrödinger equation in the spatial dimension n = 2 and quintic, defocusing nonlinear Schr¨odinger equation in the spatial dimension n = 1 with detailed calculations. We will find out on which type of equations one can use the I-method. We then mention our joint work with Engin Ba¸sako˘glu on the cubic defocusing fourth order nonlinear Schr¨odinger equation in the spatial dimension n = 4. Lastly, we discuss advantage and disadvantage of the method and share our idea of a study plan of the same equation in the spatial dimensions n = 2, 3 as future work.Schrödinger equation.ix, 45 leaves