Lecture Notes I: on Local and Global Theory for Nonlinear Schrödinger Equation

The notes serve as an introduction to the analysis of dispersive partial differential equations. They are organized as follows: • Part I focuses on basic theory for local and global analysis of the semilinear Schrödinger equation. • Part II concentrates on basic local and global theory for the Korteveg de Vries equation. • Part III gives a review of some recents results on a derivation of nonlinear dispersive equations from quantum many body systems. Dislaimer. The notes are prepared as a study tool for participants of the MSRI summer school “Dispersive Partial Differential Equations”, June 16-27, 2014. We tried to include many of the relevant references. However it is inevitable that we had to make sacrifices in the choice of the material that is included in the notes. As a consequence, there are many important works that we could not present in the notes. 1. What is a dispersive PDE Informally speaking, a partial differential equation (PDE) is characterized as dispersive if, when no boundary conditions are imposed, its wave solutions spread out in space as they evolve in time. As an example consider the linear homogeneous Schrödinger equation on the real line iut + uxx = 0, (1.1) for a complex valued function u = u(x, t) with x ∈ R and t ∈ R. If we try to find a solution in the form of a simple wave u(x, t) = Aei(kx−ωt), we see that it satisfies the equation if and only if ω = k. (1.2) The relation (1.2) is called the dispersive relation corresponding to the equation (1.1). It shows that the frequency is a real valued function of the wave number. If we denote the phase velocity by v = ωk , we can write the solution as u(x, t) = Aeik(x−v(k)t) and notice that the wave travels with velocity k. Thus the wave The work of N.P. is supported in part by NSF grant DMS-1101192. The work of N.T. is supported in part by University of Illinois Research Board Grant RB-14054. Both authors are thankful to the MSRI staff for all help in organizing the workshop. 1 2 N. PAVLOVIĆ AND N. TZIRAKIS propagates in such a way that large wave numbers travel faster than smaller ones. If we add nonlinear effects and study for example iut + uxx + |u|p−1u = 0, we will see that even the existence of solutions over small times requires delicate techniques. Going back to the linear homogenous equation (1.1), let us now consider