Nonlinear Lattice Waves

• Applicant: Professor Dr. Michael Herrmann (†)
• Subject Area: Mathematics
• Term: from 2013 to 2017
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 244645062

Many of the mathematical models used in physics, materials science, or biology are Hamiltonian lattice equations and govern the dynamics of spatially discrete media or networks of coupled oscillators. When studying such systems, coherent structures such as traveling or standing waves are of particular importance because they represent the nonlinear fundamental modes and allow us to understand how energy or information propagates through the lattice. Triggered by the rapid development in the fields of Bose-Einstein condensation, nonlinear optics, and martensitic phase transitions, nonlinear lattice waves have attracted strong attention for the last decades, but most of the rigorous results concern spatially one-dimensional systems. In our project we study Hamiltonian lattices in two or three space dimensions and plan to develop an existence theory for different types of traveling and standing waves. In this research proposal, we focus on large-amplitude waves and intend to employ optimization techniques with or without constraints to prove the existence of periodic, homoclinic, and heteroclinic waves for a broad class of nonlinear Hamiltonian lattices including atomistic models for discrete elasticity, peri-dynamical networks of coupled oscillators, and electrical transmission lattices. Moreover, comparing different models and using numerical simulations we wish to understand how geometric nonlinearities and nonlocal coupling terms affect the wave progagation in discrete media.

• DFG Programme: Research Grants