Nonlinear wave propagation in lattices

了解非线性晶格(即耦合的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基本的重要问题。其中一个主要问题涉及对特殊类型的非线性时间周期振荡的数学分析和数值计算，这些非线性时间周期振荡在许多情况下组织了动力学。特别是，空间周期波和空间局域呼吸是深入研究的对象。在这种背景下，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，而对于非光滑或非保守系统中的非线性波动的数学结果相对较少。发展非光滑或非保守系统中的非线性波的数学理论对于在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发挥作用的碰撞力学的背景下。新的分析结果可能会建议对这类系统进行新的实验。 该方案的目的是发展理论和数值工具来分析碰撞力学和非线性光学中产生的非光滑和非保守格子动力系统中的时间周期非线性波。在这种情况下，空间离散的格子模型经常遇到，特别是在多体机械系统(例如颗粒介质)或连续介质系统的有限元模型中对波的建模。该提案包括以下三个方面，这三个方面将把我的团队的研究集中在一起。 1)颗粒链中的动力学。 颗粒链是紧密堆积的弹性相互作用的颗粒集合。用带有赫兹接触力的Fermi-Pasta-Ulam晶格描述了含有不同类型粒子的一维颗粒链。我们将考虑两种不同类型的球珠在链上交替的系统，并在颗粒链中的非线性波的背景下研究约化振幅模型(例如具有对数非线性的Korteweg-de Vries方程)的性质。 2)PT对称系统的动力学。 具有增益和损耗项的PT对称晶格对于组合的奇偶变换和时间反转变换是不变的，并且被认为是类似于保守系统的行为。最近对PT对称离散非线性薛定谔方程的许多研究都涉及到定常空间局域孤子的存在性和PT对称振子有限网络中的非线性动力学问题。我们计划系统地研究无限PT-对称系统解的整体存在性，相关方程在特殊的非线性构型下的隐藏可积性，以及描述周期或局域模式的精确解的存在性。 3)谐振型非线性振子的动力学。 由于振子的多次分叉、失稳和振子振幅的共振增长，使得波在共振非线性振子中的传播变得复杂。薄壁振动机械结构(受拉的弦或固支的梁)由包含大量自由度的一维有限元模型来描述。在许多情况下，这样的格子方程可以归结为具有非光滑位势的离散Klein-Gordon方程，因为弦/梁与刚性底部之间的接触力是测量值的(对于接触时速度跳跃的回弹)或集值的(如果弦在障碍物上发生缠绕)。利用格子动力系统理论中的最新技术，我们计划重点研究这类非光滑系统驻波解的存在性和稳定性。

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.