RUI: Stability and Interaction of Coherent Structures in Lattice Differential Equations

空间扩展动力系统理论中的许多工作都致力于研究活动(或能量或质量)从空间域的一个区域传输到另一个区域的机制，如行波。霍夫曼教授将在格点微分方程(LDE)中研究相干结构的稳定性和相互作用，即指数局域行波和脉冲。这包括：(I)离散反应扩散方程中平面前锋的多维稳定性和相互作用；(Ii)哈密顿晶格中孤立波的碰撞性质；(Iii)耗散系统中相干结构的强相互作用，例如离散Fitzhugh-Nagumo方程中脉冲的湮没；(Iv)单向格点微分方程组中推前锋的稳定性，例如对流-反应方程的迎风离散或细胞神经网络模型中的推前前锋的稳定性。在晶格环境中工作的一个优点是存在和唯一性是众所周知的，因此人们可以立即转向关于动力学的更详细的问题。同时，连续的时间动力学和离散的空间结构之间的相互作用会产生有趣和微妙的现象，例如面对有理方向的锋面表现出与面对无理方向的锋面不同的行为。非线性格点微分方程通常太复杂而不能显式求解，因为根据已知函数写下了公式。然而，动力系统的方法可以用来获得不太详细的信息，这通常足以用于科学研究的目的。例如，考虑以下问题：当晶体生长时，它将接近什么形状？当合金冷却时，它的相界将如何演变？假设两个信号沿同一根光纤传播，它们将如何相互作用？入侵物种会以多大的速度侵占本土栖息地？首席研究员致力于开发数学技术来回答这些和类似的问题，用于允许空间离散相干结构的理想化物理模型。本科生将参与这项研究。

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.