Much effort in the theory of spatially extended dynamical systems has been devoted to the mechanisms, such as traveling waves, by which activity (or energy or mass) is transported from one region of the spatial domain to another. Professor Hoffman will study the stability and interaction of coherent structures, i.e. exponentially localized traveling waves and pulses, in lattice differential equations (LDE). This includes (i) multidimensional stability and interaction of planar fronts in discrete reaction diffusion equations; (ii) collision properties of solitary waves in Hamiltonian lattices such as the Fermi-Pasta-Ulam lattice; (iii) strong interaction of coherent structures in dissipative systems such as annihilation of pulses in the discrete Fitzhugh-Nagumo equation; (iv) Stability of pushed fronts in unidirectional lattice differential equations such as those obtained from upwind discretization of advection-reaction equations or in cellular neural network models. One advantage of working in the lattice setting is that existence and uniqueness is well-known, hence one can immediately turn to more detailed questions concerning the dynamics. At the same time, the interaction between the continuous temporal dynamics and the discrete spatial structure can give rise to interesting and subtle phenomenon, such as fronts facing rational directions behaving differently from fronts facing irrational directions.
Nonlinear lattice differential equations are typically too complex to solve explicitly in the sense of writing down a formula in terms of known functions. However, the methods of dynamical systems can be used to obtain less detailed information and this is often sufficient for the purposes of scientific inquiry. For example, consider the following questions: As a crystal grows, what shape will it approach? As an alloy cools, how will its phase boundaries evolve? Given two signals propagating along the same fiber, how will they interact? At what rate will an invasive species encroach upon native habitat? The Principal Investigator is concerned with developing mathematical techniques to answer these and similar questions for idealized physical models that admit spatially discrete coherent structures. Undergraduate students will participate in this research.
