Quasi-periodic waves in Fermi-Pasta-Ulam lattices

Solitary waves, i.e., localized waves that keep their shape for all times, are of practical and technological interest as they appear in many areas of physics and describe various phenomena ranging from the propagation of tsunamis to signals in optical fibers. Until the mid-nineties it was common belief that solitary waves are generic to so-called completely integrable Hamiltonian systems, e.g., the Korteweg-de Vries (KdV) equation. In the past 15 years, however, it became clear that solitary waves can also exist in large classes of non-integrable systems. Fermi-Pasta-Ulam (FPU) lattices consist of one-dimensional chains of particles coupled by an anharmonic nearest- neighbor potential and provide a model which is linked in a long wave regime to the KdV equation. FPU lattices are in general non-integrable, nevertheless, it has been shown that they have solitary wave solutions that are close to soliton solutions of the corresponding KdV problem. It is still an open question whether the many other types of permanent wave solutions of KdV persist in the non- integrable FPU context. The idea of this project is to study rational KdV solitary wave solutions and, more general, quasi-periodic KdV-like solutions on FPU lattices. An existence result for the latter could be considered a major contribution to the explanation of the so-called FPU paradox: The FPU lattice with one mode initially excited does not equilibrate within a finite time, instead all the energy returns eventually to the initially excited one. This is in contrast to Fermi`s belief that for systems with a large number of particles an anharmonic potential would generate ergodic behavior.