论文标题
通过近似理论的镜头对Pearmon世界的看法
A view of the peakon world through the lens of approximation theory
论文作者
论文摘要
山峰（峰值唯一）是某些非线性PDE所接受的特定溶液，最著名的是Camassa-Holm浅水波方程。这些解决方案采用了峰形波的形式，以粒子状的方式相互作用。在本文中，我们概述了Peamons的数学，特别强调了与分析中经典问题的联系，例如Padé近似，混合的HERMite-Padé近似，多点PADé近似，Stieltjes类型的持续分数和（BI）正交多物质。博览会遵循我们理解的时间顺序发展，探索了Camassa-Holm，Degasperis-Procesi，Novikov，Geng-Xue和Modifiend Camassa-Holm（ForQ）方程的Peamon解决方案。所有这些范式示例都是由宽松对的兼容条件引起的可集成系统，而在Peakons的背景下，重复的主题是需要从Schwartz的分布理论的意义上正确解释这些LAX对。我们通过各种近似理论问题来追踪从分布宽松对到Pearmon溶液的明确公式的路径，我们用图形说明了Pearmon动力学。
Peakons (peaked solitons) are particular solutions admitted by certain nonlinear PDEs, most famously the Camassa-Holm shallow water wave equation. These solutions take the form of a train of peak-shaped waves, interacting in a particle-like fashion. In this article we give an overview of the mathematics of peakons, with particular emphasis on the connections to classical problems in analysis, such as Padé approximation, mixed Hermite-Padé approximation, multi-point Padé approximation, continued fractions of Stieltjes type and (bi)orthogonal polynomials. The exposition follows the chronological development of our understanding, exploring the peakon solutions of the Camassa-Holm, Degasperis-Procesi, Novikov, Geng-Xue and modified Camassa-Holm (FORQ) equations. All of these paradigm examples are integrable systems arising from the compatibility condition of a Lax pair, and a recurring theme in the context of peakons is the need to properly interpret these Lax pairs in the sense of Schwartz's theory of distributions. We trace out the path leading from distributional Lax pairs to explicit formulas for peakon solutions via a variety of approximation-theoretic problems, and we illustrate the peakon dynamics with graphics.
