Physica D: Nonlinear Phenomena ( IF 4 ) Pub Date : 2020-05-05 , DOI: 10.1016/j.physd.2020.132538 Timothy E. Faver , Hermen Jan Hupkes
The diatomic Fermi–Pasta–Ulam–Tsingou (FPUT) lattice is an infinite chain of alternating particles connected by identical nonlinear springs. We prove the existence of micropteron traveling waves in the diatomic FPUT lattice in the limit as the ratio of the two alternating masses approaches 1, at which point the diatomic lattice reduces to the well-understood monatomic FPUT lattice. These are traveling waves whose profiles asymptote to a small periodic oscillation at infinity, instead of vanishing like the classical solitary wave. We produce these micropteron waves using a functional-analytic method, originally due to Beale, that was successfully deployed in the related long wave and small mass diatomic problems. Unlike the long wave and small mass problems, this equal mass problem is not singularly perturbed, and so the amplitude of the micropteron’s oscillation is not necessarily small beyond all orders (i.e., the traveling wave that we find is not necessarily a nanopteron). The central challenge of this equal mass problem hinges on a hidden solvability condition in the traveling wave equations, which manifests itself in the existence and fine properties of asymptotically sinusoidal solutions (Jost solutions) to an auxiliary advance–delay differential equation. The novelty compared to previous approaches is that this operator is neither a Fourier multiplier nor a small nonlocal perturbation of a classical differential operator. This causes fundamental technical obstructions, which we overcome by developing new functional-analytic techniques to uncover the asymptotic phase shifts of the Jost solutions.
中文翻译:
在相同质量极限下双原子费米-帕斯塔-乌拉姆-津古格晶格中的微型p行波
双原子费米-帕斯塔-乌拉姆-津古(FPUT)晶格是由相同的非线性弹簧连接的交替粒子的无限链。我们证明了当两个交替质量的比率接近1时,在双原子FPUT晶格中存在极限的微滤子行波,此时双原子的晶格还原为易于理解的单原子FPUT晶格。这些是行波,其轮廓渐近于无穷远处的小周期振荡,而不是像传统的孤立波一样消失。我们使用功能分析方法产生这些微p波,最初是由于Beale所致,该方法已成功应用于相关的长波和小质量双原子问题。与长波和小质量问题不同,该等质量问题不会受到单独的干扰,因此,微光子振荡的振幅不一定会超出所有阶数(即,我们发现的行波不一定是纳升子)。等质量问题的主要挑战在于行波方程中的隐性可溶性条件,这本身表现为辅助超前-时滞微分方程的渐近正弦解(Jost解)的存在和优良性质。与以前的方法相比,新颖之处在于该算子既不是傅立叶乘数,也不是经典微分算子的小局部扰动。这导致了基本的技术障碍,我们通过开发新的功能分析技术来发现Jost解的渐近相移,从而克服了这些障碍。