Discrete Solitons: Methods, Theory and Applications

University Of Massachusetts Amherst, Amherst MA

Investigators

Abstract

The aim of this proposal is to use a number of mathematical tools and techniques (Hamiltonian normal forms, variational analysis, exponential asymptotics, integrability methods, multiscale techniques, homogenization and Evans function methods among others), in conjunction with numerical methods (continuation and bifurcation theory tools, Newton type methods together with numerical linear stability analysis and direct time integration) to systematically explore nonlinear waves in discrete systems. In particular, we plan to address the following aspects of the behavior, dynamics and stability of the waves in lattice settings: (1) The role of spatial dimensionality on nonlinear waves. Most of the earlier studies had been conducted in 1+1 (1 space, 1 time) dimension. We can now go even in 3+1 dimensions and examine physically realistic settings. (2) The interplay between disorder, nonlinearity and discreteness. It is well known that disorder can induce localization. Understanding the interplay of this mechanism with nonlinearity, especially in realistic discrete settings is then crucial, as defects are ubiquitous in physical systems. (3) Travelling waves in discrete systems are also of paramount importance. Intrinsic Localized Modes (ILM's) have the intriguing property of bottlenecking the energy. But could they carry it over (even more so in a targeted way) from one molecule to another (from one lattice site to another)? If so, they would be natural candidates for many bioenergetics processes, such as photosynthesis. (4) The study of instabilities of such waves. We believe that we are now close to a general classification of the possible instabilities and to a connection of these with the underlying symmetries of the physical problem. (5) Finally, the comprehension of progressively more complex physical models is also of interest. The latter involve additional physical perturbations such as, for example, long range interactions, boundary conditions, the interaction of multiple waves between them and with defects. Intrinsic Localized Modes (ILM's) have been a topic of increasing focus over the past decade as their role in energy localization and transport has been appreciated in a variety of contexts. Their applications span nonlinear optics and telecommunications (optical fibers and waveguides), atomic physics (BEC, an issue of fundamental relevance as highlighted by the Physics Nobel Prize in 2001 awarded for its experimental observation), condensed matter physics (superconductivity and charge density waves), biophysics (the local breaking of DNA and conformational changes in proteins), and environmental science (nucleation of liquid droplets in the atmosphere). The areas of interest are diverse and broad, the impact of the understanding of the fundamental physics is potentially very deep, but the underlying mathematical principles are simple and unifying. These models share a common structure of nonlinear complex behavior, the spatio-temporal variation of which we wish to explore. Understanding this behavior and the role of (nonlinear) waves in it is of fundamental interest in all these fields. The nonlinear waves represent the electric field of light in optics, the quantum-mechanical Bose particle wavefunction in BEC, the DNA base-pair distance or the liquid-vapor interface of a nucleating droplet in the atmosphere. The properties of these waves, their stability, dynamical evolution and internal structure are therefore at the heart of a wealth of physical effects. The goal of our research is to explore these features using a combination of analytical and computational mathematical techniques and physical intuition. Our project addresses, in particular, lattice dynamical systems, where space is discrete, as is the case in many fundamental applications, such as optical waveguides, DNA, and arrays of superconducting Josephson junctions.