In this thesis, Fabrice Bethuel, Philippe Gravejat, and Jean-Claude Saut discussed the existence of travelling wave solutions to the Gross-Pitaevskii equation in RN , where N = 2, 3. c∂1u + ∆u + u(1 − |u|2) = 0 In the first four sections, we survey the theorems based on minimizing energy under momentum constraint. Also, we give some motivations about how the theorems are constructed. In the final section, we discuss the future works of Gross-Pitaevskii equation.
