Nonlinear wave propagation in lattices

理解非线性晶格（即耦合非线性振荡器的大型网络）的动力学是力学，光学，凝聚态物理和生物学中的一个基本重要问题。其中一个主要的问题涉及的数学分析和数值计算的特殊类别的非线性时间周期振荡，组织在许多情况下的动态。特别是，空间周期波和空间局部呼吸是深入研究的对象。在这方面，许多理论和数值工作都集中在光滑和保守的非线性系统，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的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。新的分析结果可能会建议用这种系统进行新的实验。 该提案的目的是开发理论和数值工具，用于分析由冲击力学和非线性光学产生的非光滑和非保守晶格动力学系统中的时间周期非线性波。空间离散格点模型在这方面经常遇到，特别是在多体力学系统（如颗粒介质）或连续系统的有限元模型中的波建模。该提案包括以下三个方面，这将集中在我的团队的研究。 1)颗粒链中的动力学。 颗粒链是弹性相互作用粒子的紧密堆积的集合。用具有赫兹接触力的费米-帕斯塔-乌拉姆格点描述含有不同类型颗粒的一维颗粒链。我们将考虑一个系统，两种不同类型的球珠交替链和研究属性的约化振幅模型（如Korteweg-de弗里斯方程与对数非线性）的背景下，非线性波的颗粒链。 2)PT对称系统中的动力学。 PT-对称晶格的增益和损失的条款是不变的组合奇偶和时间反转变换，并表现出类似的保守系统。PT对称离散非线性薛定谔方程的许多研究涉及到PT对称振子有限网络中定态空间局域孤子的存在性和非线性动力学。我们计划系统地研究无穷PT-对称系统解的整体存在性，特殊非线性配置的相关方程的隐可积性，以及描述周期或局部模式的精确解的存在性。 3)共振非线性振子动力学。 波在共振非线性振子中的传播由于振子的分叉、稳定性损失和振幅的共振增长而变得复杂。薄振动机械结构（张力下的弦或固支梁）由包含大量自由度的一维有限元模型描述。在许多情况下，这样的晶格方程可以简化为具有非光滑势的离散克莱因-戈登方程，因为弦/梁和刚性底部之间的接触力是测量值（对于在接触时间具有速度跳跃的反弹）或集值（如果弦缠绕在障碍物上）。利用最近的技术从理论的格子动力系统，我们计划集中在驻波解的存在性和稳定性，在这样的非光滑系统。

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.