Nonlinear wave propagation in lattices

理解非线性晶格（即耦合非线性振子的大型网络）的动力学在力学、光学、凝聚态物理和生物学中是一个至关重要的问题。其中一个主要问题涉及到在许多情况下组织动力学的特殊类别的非线性时间周期振荡的数学分析和数值计算。特别是空间周期波和空间局域呼吸是深入研究的对象。在这种情况下，许多理论和数值工作都集中在光滑和保守的非线性系统上，而相对较少的数学结果可用于非光滑或非保守系统中的非线性波。发展非光滑或非保守系统中非线性波的数学理论对于许多应用中的建模目的是重要的，特别是在单边接触和摩擦起作用的冲击力学的背景下。新的分析结果可能建议用这种系统进行新的实验。

Understanding the dynamics of nonlinear lattices (i.e. large networks of coupled nonlinear oscillators) is a problem of fundamental importance in mechanics, optics, condensed matter physics, and biology. One of the major issues concerns the mathematical analysis and numerical computations of special classes of nonlinear time-periodic oscillations that organize the dynamics in many situations. In particular, spatially periodic waves and spatially localized breathers are the objects of intensive research. In this context, many theoretical and numerical works have focused on smooth and conservative nonlinear systems, whereas relatively few mathematical results are available for nonlinear waves in nonsmooth or nonconservative systems. Developing the mathematical theory of nonlinear waves in nonsmooth or nonconservative systems is important for modelling purposes in many applications, in particular in the context of impact mechanics where unilateral contacts and friction come into play. New analytical results may suggest new experiments with such systems. The aim of the proposal is to develop theoretical and numerical tools for the analysis of time-periodic nonlinear waves in nonsmooth and nonconservative lattice dynamical systems arising from impact mechanics and nonlinear optics. Spatially discrete lattice models are frequently encountered in this context, in particular for the modeling of waves in many-body mechanical systems (e.g. granular media) or in finite element models of continuum systems. The proposal consists of the following three areas, which will group together the research of my team. 1) Dynamics in granular chains. Granular chains are closely packed ensembles of elastically interacting particles. One-dimensional granular chains with different types of particles are described by the Fermi-Pasta-Ulam lattice with Hertzian contact forces. We shall consider a system with two different types of spherical beads alternating on the chain and study properties of the reduced amplitude models (such as the Korteweg-de Vries equation with logarithmic nonlinearity) in the context of nonlinear waves in granular chains. 2) Dynamics in PT-symmetric systems. PT-symmetric lattices with gain and loss terms are invariant with respect to combined parity and time reversal transformations and are seen to behave similar to the conservative systems. Many recent studies of the PT-symmetric discrete nonlinear Schrodinger equation concern with the existence of stationary spatially localized solitons and the nonlinear dynamics in finite networks of PT-symmetric oscillators. We plan to systematically study global existence of solutions in the infinite PT-symmetric systems, hidden integrability of the relevant equations for special nonlinear configurations, and existence of exact solutions describing periodic or localized modes. 3) Dynamics in resonant nonlinear oscillators. Wave propagation in resonant nonlinear oscillators becomes complicated because of a number of bifurcations, loss of stability, and resonant growth of the amplitudes of oscillators. Thin oscillating mechanical structures (a string under tension or a clamped beam) are described by a one-dimensional finite-element model involving a large number of degrees of freedom. In many cases, such lattice equations can be reduced to the discrete Klein-Gordon equations with nonsmooth potentials because the contact force between the string/beam and the rigid bottom is measure-valued (for rebounds with velocity jumps at contact times) or set-valued (if a wrapping of the string on the obstacle occurs). Using recent techniques from the theory of lattice dynamical systems, we plan to focus on the existence and stability of standing wave solutions in such nonsmooth systems.