************************************************************ London Dynamical Systems Group ************************************************************ LDSG Workshop on Hamiltonian Lattice Dynamics Wednesday 16 November 2005, Imperial College London, 180 Queens Gate Huxley Building, Room 140 Program: 2:00 - 2:50 pm Jonathan Wattis (Nottingham) Moving breathers in one- and two-dimensional Fermi-Pasta-Ulam lattices 2:50 - 3:00 pm: small break 3:00 - 3:50 pm: Johannes Zimmer (Bath) Solitary waves in Nonconvex Fermi-Pasta-Ulam lattices 3:50 - 4:30 pm: coffee break 4.30 - 5:20 pm: Bob Rink (Imperial) A family of FPU lattices with long time integrable evolution of long waves. 5:20 - .... drinks in the Queen's arms Abstracts are included below. LDSG is a joint research network of the dynamical systems groups at Imperial College London, Queen Mary, University College London and the University of Surrey, and financially supported by the London Mathematical Society. ************************************************************ London Dynamical Systems Group http://www.ma.imperial.ac.uk/~jswlamb/LDSG DynamIC: Dynamical systems at Imperial College London http://www.ma.imperial.ac.uk/DynamIC ************************************************************ Abstract of "Moving breathers in one- and two-dimensional Fermi-Pasta-Ulam lattices" by Jonathan Wattis: We use multiple scales asymptotic techniques to obtain approximations to moving breather solutions in one- and two-dimensional Fermi-Pasta-Ulam (FPU) lattices. In one-dimensional chains we find a family of solutions which can be continuously varied from traditional breathers to breathing-kink travelling waves. We study interaction potentials with both cubic and quartic contributions, and determine an inequality on the cubic and quartic coefficients which must be satisfied for breathers to exist. The breathing-kink solutions have a kink amplitude which can be arbitrarily small (in contrast with classical kinks in the quartic FPU, which have a nonzero minimum amplitude). In two-dimensional lattices, similar asymptotic analysis yields an ellipticity criterion for the existence of breathers. Although a 2D NLS equation is derived, which is known to exhibit soliton collapse and an unstable breather mode, we show that higher-order terms from the asymptotic calculations can stabilise the breather. Numerical simulations show the long-lived nature of breathers in both one- and two-dimensional systems. Abstract of "Solitary waves in Nonconvex Fermi-Pasta-Ulam lattices" by Johannes Zimmer: Travelling waves in a one-dimensional chain of atoms will be investigated. The aim is to allow for nonconvex energy densities, which occur in the theory of phase transforming solids, such as martensitic crystals. The existence of solitary waves with a prescribed asymptotic strain will be shown under certain assumptions on the asymptotic strain and the wave speed. Connections to previous results will be discussed. This is joint work with Hartmut Schwetlick (Bath). Abstract of "A family of FPU lattices with long time integrable evolution of long waves" by Bob Rink: It is well-known that the integrable Korteweg-de Vries equation can be derived as a first order asymptotic equation describing the evolution of long waves in the Fermi-Pasta-Ulam lattice. On the other hand, numerical experiments indicate that depending on the exact nonlinearity of the inter-particle forces, the long-time behavior of its solutions may display features of nonintegrability and chaos. We investigate this phenomenon by computing a second order asymptotic approximation beyond the KdV equation and we show that the FPU lattice has an integrable second order asymptotic approximation, namely a KdV5 equation, if and only if its nonlinearities satisfy a codimension-one condition. Remarkably, the integrable Toda-lattice is not part of this family.