Nonlinear stability of periodic waves in dissipative-dispersive systems (A14*)

• Subject Area: Mathematics
• Term: since 2023
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 258734477

Our plan is to establish nonlinear stability of periodic waves against localized perturbations in various dissipative-dispersive or purely dispersive systems. For many paradigm examples of such systems, such as the Klein-Gordon, Korteweg-de Vries, or nonlinear Schrödinger equations, existence and spectral stability of periodic waves are well-established, whereas nonlinear stability against localized perturbations is still a largely unresolved problem. We adopt an approach that combines spatio-temporal phase modulation with iterative estimates on the Duhamel formulation and has been successfully applied to a wide range of dissipative models. We aim to employ the space-time resonances method to uncover dispersive decay in this Duhamel-based approach. Thereby, we extend its applicability to purely dispersive models or to dissipative-dispersive systems, where dispersive effects are decisive for nonlinear stability. Examples of such dissipative-dispersive systems are coupled equations of the Ginzburg-Landau type arising in nonlinear fiber optics.

• DFG Programme: Collaborative Research Centres
• Subproject of: SFB 1173: Wave phenomena: analysis and numerics
• Applicant Institution: Karlsruher Institut für Technologie
• Project Heads: Professorin Dr. Dorothee Frey; Dr. Björn de Rijk