Nonlinear Lattice Waves
Final Report Abstract
The most important coherent modes in nonlinear Hamiltonian lattice equations are traveling waves, which can be regarded as the fundamental modes of energy transport in spatially discrete media and provide the building blocks for more complex solutions. From mathematical point of view, lattice waves are solutions to nonlinear advance-delay-differential equations, for which a complete mathematical theory has not yet been developed. In this project we studied solitary waves in one- and two-dimensional Fermi-Pasta-Ulam-Tsingou lattices, where most of our work concerns either one of two complementary asymptotic regimes. The waves in the Korteweg-deVries limit propagate with near sonic speed, have large wave numbers, and carry low energies. A rigorous mathematical theory concerning the existence, uniqueness, and orbital stability of those waves has been developed by Friesecke and Pego but was restricted to one-dimensional chains with nearest neighbor interactions. In this project we first generalized the existence result to chains with nonlocal interactions, which required us to develop a more robust asymptotic framework. In a second step we investigated near sonic waves in two-dimensional lattices and identified for the first time existence criteria that cover many lattice geometries and interaction potentials without any restriction on the propagation direction. From a mathematical point of view, our work applies singular perturbation techniques to a certain class of nonlinear and nonlocal eigenvalue problems in vector-valued function spaces. Traveling waves in the high-energy limit move very fast but remain localized on the lattice. Our first contribution to the corresponding mathematical theory are explicit approximation formulas for the solutions to the underlying advance-delay-differential equation in chains with singular potential. In particular, using taylor-made multi-scale arguments we proved that the global properties of high-energy waves are completely determined by a local shape ODE as well as a hierarchy of differential relations and matching conditions. Our second result on high-energy waves guarantees their dynamical stability in the nonlinear and orbital sense of the Friesecke-Pego theory. The mathematical proof exploits the properties of the linearized shape ODE and relies on a careful multi-scale analysis of linear advancedelay-differential operators with space-dependent but singular coefficient functions.
Publications
- Stability of high-energy solitary waves in Fermi-Pasta-Ulam-Tsingou chains
M. Herrmann, K. Matthies
- Asymptotic formulas for solitary waves in the high-energy limit of FPU-type chains, Nonlinearity, vol. 28/8, pp. 2767-2789, 2015
M. Herrmann, K. Matthies
(See online at https://doi.org/10.1088/0951-7715/28/8/2767) - KdV waves in atomic chains with nonlocal interactions, Discrete and Continuous Dynamical Systems - Series A, vol. 34/4, pp. 2047-2067, 2016
M. Herrmann, A. Mikikits-Leitner
(See online at https://doi.org/10.3934/dcds.2016.36.2047) - High-energy waves in superpolynomial FPU-type chains, Journal of Nonlinear Science, vol. 27/1, pp. 213-240, 2017
M. Herrmann
(See online at https://doi.org/10.1007/s00332-016-9331-8) - Traveling waves in two-dimensional FPU lattices, PhD Thesis, Institute for Computational and Applied Mathematics, University of Münster, 2017
F. Chen
- Uniqueness of solitary waves in the high-energy limit of FPU-type chains, in Patterns of Dynamics, Springer, 2017
M. Herrmann, K. Matthies
(See online at https://doi.org/10.1007/978-3-319-64173-7_1) - KdV-like solitary waves in two-dimensional FPU-lattices, Discrete and Continuous Dynamical Systems - Series A, vol. 38/5, pp. 2305–2332, 2018
F. Chen, M. Herrmann
(See online at https://doi.org/10.3934/dcds.2018095)
