From stability of solitons to global behavior: A unified study of nonlinear Schrödinger and Euler–Korteweg equations with non-trivial far-field

• Applicant: Dr. Jordan Berthoumieu
• Subject Area: Mathematics
• Term: since 2026
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 577909696

This project focuses on a unified study of nonlinear Schrödinger (NLS) and Euler–Korteweg (EK) equations with non-trivial far-field conditions, combining dispersive analysis and hydrodynamic approaches. These models are central to nonlinear optics, Bose–Einstein condensates, superfluidity, and quantum plasmas. NLS equations with non-vanishing conditions at infinity display much richer dynamics than in the classical vanishing case, notably the emergence of coherent structures such as dark solitons. The applicant’s previous results established existence, uniqueness, and orbital/asymptotic stability of traveling waves in this non-integrable setting. Through the Madelung transform, NLS connects to Euler–Korteweg systems, providing a bridge to quantum fluids and allowing the study of transverse stability and instability phenomena. The main objectives are: -Collisions of solitons in 1D (NLS): investigate asymptotic stability of chains of solitons and analyze collisions (elastic or inelastic) between traveling waves in non-integrable systems. -The multidimensional Cauchy problem (NLS): deepen the understanding of global existence vs. blow-up for general nonlinearities, especially focusing cases and higher dimensions. The goal is to broaden recent results obtained in the defocusing framework. -Transverse stability for (EK): study the transition in stability of traveling waves from one to higher dimensions, particularly on product spaces and determine critical thresholds for stability/instability.

• DFG Programme: Position