Qualitative Properties of Dispersive PDEs

This book provides a valuable collection of contributions by distinguished scholars presenting the state of the art and some of the most significant latest developments and future challenges in the field of dispersive partial differential equations. The material covers four major lines: (1) Long time behaviour of NLS-type equations, (2) probabilistic and nonstandard methods in the study of NLS equation, (3) dispersive properties for heat-, Schrödinger-, and Dirac-type flows, (4) wave and KdV-type equations. Across a variety of applications an amount of crucial mathematical tools are discussed, whose applicability and versatility goes beyond the specific models presented here. Furthermore, all contributions include updated and comparative literature.

Part I: Long-time behavior of NLS-type equations.- 1 Scipio Cuccagna, Note on small data soliton selection for nonlinear Schrödinger equations with potential.- 2 Jacopo Bellazzini and Luigi Forcella, Dynamics of solutions to the Gross-Pitaevskii equation describing dipolar Bose-Einstein condensates.- Part II: Probabilistic and nonstandard methods in the study of NLS equations.- 3 Renato Luca, Almost sure pointwise convergence of the cubic nonlinear Schrödinger equation on T^2.- 4 Nevena Dugandžija and Ivana Vojnovic, Nonlinear Schrödinger equation with singularities.- Part III: Dispersive properties.- 5 Vladimir Georgiev, Alessandro Michelangeli, Raffaele Scandone, Schrödinger flow's dispersive estimates in a regime of re-scaled potentials.- 6 Federico Cacciafesta, Eric Sere, Junyong Zhang, Dispersive estimates for the Dirac-Coulomb equation.- 7 Matteo Gallone, Alessandro Michelangeli, Eugenio Pozzoli, Heat equation with inverse-square potential of bridging type across twohalf-lines.- Part IV: Wave and Kdv-type equations.- 8 Felice Iandoli, On the Cauchy problem for quasi-linear Hamiltonian KdV-type equations.- 9 Vladimir Georgiev and Sandra Lucente, Linear and nonlinear interaction for wave equations with time variable coefficients.- 10 Matteo Gallone and Antonio Ponno, Hamiltonian field theory close to the wave equation: from Fermi-Pasta-Ulam to water waves.

Vladimir Georgiev is a former Alexander von Humboldt fellow. He is a Full Professor of mathematics at the University of Pisa. The main fields of the research interests involve decay estimates for equations of Mathematical Physics on flat or curved space - time, smoothing and Strichartz estimates for evolution problems, global existence of small and large data solutions to equations of classical quantum mechanics, existence and stability of solitary waves, Maxwell–Dirac and Maxwell–Scrödinger equation, scattering and long range effects for relativistic and non – relativistic particles and fields.

Alessandro Michelangeli is an Alexander von Humboldt Senior Researcher at the Institute for Applied Mathematics and at the Hausdorff Center for Mathematics, Bonn, and a member of the Institute of Theoretical Quantum Technologies Trieste. He also held positions at the LMU Munich and the SISSA Trieste. His research is at the interface of analysis, mathematical physics,and theoretical physics, with expertise in functional analysis, operator theory, spectral theory, non-linear partial differential equations, and quantum mechanics.

Raffaele Scandone is a postdoctoral researcher at Gran Sasso Science Institute, Italy. He received his PhD in Mathematics at SISSA, Italy, in 2014. His research interests lie in the area of dispersive PDEs, with a particular focus on Schrodinger-type equations and quantum hydrodynamics.