• evolution problem
• ground states
• Schrödinger

In this work of thesis, we introduced a new problem, which we have not seen being studied in the literature. The problem consists of a one-dimensional dissipative non-linear Schrödinger equation, which could arise for example in non-linear fiber-optic problems. Firstly, we investigated the existence of ground states, wave-solutions of minimal energy. We were able to prove existence results using the concentrated compactness principle introduced by P.L. Lions. Secondly, we focused on the time-evolution problem. For this topic, we had to introduce the stationary phase method in order to obtain Strichartz dispersive estimates. Thanks to this we were able to prove local and global existence, given suitable initial datums. Throughout the thesis generalizations to higher dimensions were given. Lastly, we tried to introduce the problem of stability of ground states to Lyapunov's stability to orbital stability, underlining the difficulties of such theory and leaving some open windows for future works.