Nonlocal nonlinear dynamics
Résumé
It is nowadays well appreciated that nonlocality has a dramatic impact on the dynamics in nonlinear systems. Nonlocality in spatial domain is usually associated with transport processes or inter-molecular long-range interactions. Nonlocality in temporal domain, i.e., non-instantaneity, is intrinsically tied to dispersive properties of the nonlinear response. After a brief introduction to nonlocal nonlinear systems, with special emphasize on similarities and differences between spatial and temporal nonlocality, I will first focus on wave collapse in systems with attractive spatially nonlocal nonlinarities. I will present rigorous collapse criteria for systems with singular nonlocal nonlinear interaction, similar to those known for the famous local nonlinear Schroedinger equation. In the second part of my talk I will switch to a more applied subject involving temporal nonlocality, namely material processing with intense fs laser pulses. Material modifications are induced via generation of an electron plasma, a highly non-instantaneous process. Understanding the complicated nonlinear spatio-temporal propagation dynamics of the driving pulse turns out to be crucial to optimize the energy deposition in the material. Examples relevant to fs-surgery in ophthalmology are finally discussed.