Dispersive partial differential equations. Wellposedness and applications (Q2814158)

The book is a manual for beginning graduate students in the field of the general theory of nonlinear partial differential equations. The material is presented in the rigorous mathematical style, providing proofs of formal theorems, rather than less strict considerations which may be often encountered in physics literature. The attention is focused on one-dimensional equations, especially the nonlinear Schrödinger and Korteweg - de Vries equations, with periodic boundary conditions, although the well-known exact integrability of these equations, i.e., the inverse-scattering transform, is not used. Main topics addressed in the book are the proof of the well-posedness of the considered problems (with periodic boundary conditions) and global existence of solutions. In particular, application of the smoothing method are considered in detail. As an application, the theory of the nonlinear Talbot effect (periodic reappearance of sharp grating profiles in light fields transmitted through a grating) is presented.