Inhomogeneous Nonlinear Schrödinger Equations

The inhomogeneous nonlinear Schrödinger equation appears in a variety of physical settings, for example, in nonlinear optical systems with spatially dependent interactions. The wide range of applications and intriguing analytical difficulties have led to an extensive mathematical study of the inhomogeneous nonlinear Schrödinger equation in recent years. The purpose of this book is to give an introduction to the recent developments in the study about the local and global well-posedness as well as scattering and blow-up for the inhomogeneous nonlinear Schrödinger equation. First, we establish the local well-posedness as well as the small data global well-posedness for the inhomogeneous nonlinear Schrödinger equation in the fractional Sobolev spaces. Second, we study the continuous dependence for the inhomogeneous nonlinear Schrödinger equation in the standard sense in the fractional Sobolev spaces. Finally, the global existence and blow-up results in the energy space for the inhomogeneous nonlinear Schrödinger equation with inverse-square potential are presented.