Global existence of solutions for a fourth-order nonlinear Schrödinger equation

In this work we study the Cauchy problem of a fourth-order nonlinear Schrödinger equation which arises from certain physical applications. We consider only the cases n=1,2,3n=1,2,3. Local existence of solutions for initial data belonging to Sobolev spaces with index greater than n/2n/2 is established by using the standard contraction mapping argument. The main topic is proving that the solution is global if either the exponent of the nonlinear term is sub-critical or it is critical or super-critical but the initial data are small. This result extends the corresponding result of Fibich et al. obtained in 2002 to the super-critical case and to a more general equation. The analysis is based on applications of conservation laws for this equation.