The initial value problem for nonlinear Schrödinger equations

Abstract

I will review some recent work done in collaboration with C. E. Kenig, G. Ponce and C. Rolvung on a general method to solve locally in time the initial value problem for non-linear Schrödinger equations under some natural hypotheses of decay and regularity of the coefficients. Also some non-trapping conditions of the solutions of the hamiltonian flow associated to the initial data is needed. We will not assume ellipticity on the matrix of the leading order coefficients but just a non-degeneracy condition. The method is based on energy estimates which can be performed thanks to the construction of an integrating factor. This construction is of independent interest and relies on the analysis of some new pseudo-differential operators.