Standing and Outward Radiating Wave Solutions of Nonlinear Helmholtz Equations

• Applicant: Professor Dr. Tobias Weth
• Subject Area: Mathematics
• Term: from 2014 to 2019
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 263284198

Wave propagation in nonlinear media has emerged as one of the most intriguing research topics in the field of nonlinear analysis and partial differential equations. Whereas the occurence of standing waves, dispersion, and wave scattering is by now reasonably well understood in a linear context, it gives rise to an abundance of open questions in the nonlinear setting. The present project is devoted to the study of time periodic solutions of the nonlinear Klein-Gordon equation representing standing or scattered outward radiating waves with high frequency. The time periodic ansatz leads to the nonlinear Helmholtz equation, which has attracted growing attention recently as it arises in various problems where the presence of essential spectrum hampers the understanding of nonlinear effects. The present proposal builds on the progress made in the first funding period, in which results on the existence of standing wave solutions and solution continua have been derived, for non-critically growing nonlinearities, by means of dual variational and topological fixed point methods. We shall now tackle challenging open questions concerning the impact of critical exponents, the shape of dual ground states, the occurence of concentration phenomena, the contination of solution branches, and the nature of stationary scattering.

• DFG Programme: Research Grants