Peakon integrable nonlinear equations and related approximation problems: the distributional approach.

Peakon 可积非线性方程和相关逼近问题：分布方法。

• 批准号: RGPIN-2019-04051
• 负责人: Szmigielski, Jacek
• 金额: $ 1.53万
• 依托单位: University of Saskatchewan
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2022
• 资助国家: 加拿大
• 起止时间: 2022-01-01 至 2023-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=759142
• 关键词: Peakon; integrable; nonlinear; equations; related

Integrable partial differential equations arose as approximate model equations in the mathematical description of wave phenomena in fluids, plasmas and optical fibres. They are special in that they possess infinitely many integrals of motion, a sign of a rich mathematical structure. P. Lax pointed out at the very early stage of research into this special class of partial differential equations that the underlying reason for the existence of infinitely many conserved quantities is that one can associate with each of these equations a linear operator which undergoes an isospectral (spectrum preserving) deformation responsible for the existence of a large family of constants of motion. This point of view eventually resulted in the formulation of a paradigm of integrable systems, which puts the so called Lax pairs front and centre of the theory. This point of view has its geometric counterpart in a theory of flat connections, as well as, in a symplectic approach based on the presence of bi-Hamiltonian structures. The current proposal focuses on the first interpretation, using Lax pairs and their isospectral deformations, with one significant generalization: the Lax pairs are distributional (Schwartz) Lax pairs. The main impetus for this line of research comes from a class of exciting integrable models discovered over last two decades which exhibit several novel features, the central of which is the existence of localized coherent modes, called peakons, with singular behaviour in the spatial derivative of the profile.  The peakon equations result from a careful consideration of distributional Lax pairs leading to interesting and unexpected results, i.e. forcing a particular type of multiplication rules for certain classes of distributions with singular support, leading to a relevant extension of "Lax integrable" equations into the domain of significant non-smoothness for which the classical Lax formalism fails. The general objectives of the proposed program are: (1) to establish a complete theory of isospectral deformations of the n-th order inhomogeneous strings (n=2, n=3, n=4, corresponding to a classical string, a cubic string, an Euler-Bernoulli  beam, respectively) and the role of, possibly generalized, Cauchy biorthogonal polynomials for n greater than 3. This, in particular, would address the inverse problem for the Euler-Bernoulli beam which is of significance elsewhere in view of its relevance to the geophysical inverse problem;  (2) to classify peakon distributional Lax pairs and pertinent boundary value problems;  (3) to build a theory of mixed Hermite-Pade approximations of the so called Nikishin systems, generalizing earlier results on the Nikishin systems with two measures originating in the cubic string problem formulated by the applicant for the Degasperis-Procesi equation and establish a link between these approximations and inverse problems for n-order inhomogeneous strings.

P. Lax 指出，可积偏微分方程是作为流体、等离子体和光纤中波动现象的数学描述中的近似模型方程而出现的，它们的特殊之处在于它们具有无限多个运动积分，这是丰富的数学结构的标志。在对这一类特殊偏微分方程研究的早期阶段，存在无限多个守恒量的根本原因是，人们可以将这些方程中的每一个与一个经历同谱（谱这种观点最终导致了可积系统范式的制定，该范式将所谓的 Lax 对置于理论的前沿和中心。在平面连接理论中，以及在基于双哈密尔顿结构存在的辛方法中，当前的提议侧重于第一种解释，使用 Lax 对及其等谱变形，其中一个重要的解释是。泛化：Lax 对是分布（Schwartz）Lax 对 这一系列研究的主要动力来自于过去二十年发现的一类令人兴奋的可积模型，这些模型表现出一些新颖的特征，其核心是局部相干的存在。称为 Peakons 的模式，在轮廓的空间导数中具有奇异行为 Peakon 方程是通过仔细考虑分布 Lax 对而得出的，从而产生有趣且意想不到的结果，即对某些特定类型的乘法规则强制执行。具有奇异支持的分布类别，导致“Lax 可积”方程相关扩展到经典 Lax 形式主义失败的显着非光滑域。 所提议程序的总体目标是：（1）建立完整的非光滑域。 n阶非齐次弦的等谱变形理论（n=2、n=3、n=4，对应经典弦、立方弦、欧拉-伯努利n 大于 3 时的柯西双正交多项式的作用，以及可能广义的柯西双正交多项式的作用。特别是，这将解决欧拉-伯努利梁的反问题，鉴于其与地球物理的相关性，该问题在其他地方具有重要意义。逆问题；(2) 对 Peakon 分布 Lax 对和相关边值问题进行分类；(3) 建立该问题的混合 Hermite-Pade 近似理论；称为 Nikishin 系统，利用申请人为 Degasperis-Procesi 方程提出的三次弦问题中的两个度量来概括 Nikishin 系统的早期结果，并在这些近似值和 n 阶非齐次弦的逆问题之间建立联系。