Propriedades das soluções de uma equação de Schrödinger não-linear de alta ordem
Nonlinear Schródinger equation
We study properties of solutions for a higher order nonlinear Schrodinger equation. When the coefficients appearing in the linear terms of the equation are smooth functions of time, we establish local well-posedness for the associated initial value problem (IVP) with data in Sobolev spaces of order greater or equal than one fourth. We also consider the equation with constant coefficients and show that the local solutions of the IVP can be extended globally in Sobolev spaces of index greater than five ninth. Other problem we consider here is related to unique continuation principles. In particular, we prove that solutions of the IVP with compact support in two different times have to be zero. Finally, we investigate whether or not the results obtained for the IVP are the best possible and ill-posedness issues for the problem.