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。

基本信息

  • 批准号:
    RGPIN-2019-04051
  • 负责人:
  • 金额:
    $ 1.53万
  • 依托单位:
  • 依托单位国家:
    加拿大
  • 项目类别:
    Discovery Grants Program - Individual
  • 财政年份:
    2020
  • 资助国家:
    加拿大
  • 起止时间:
    2020-01-01 至 2021-12-31
  • 项目状态:
    已结题

项目摘要

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.
在流体,等离子体和光纤中的波浪现象中,可集成的部分微分方程作为近似模型方程式。在对这种特殊类别方程的研究的早期阶段,绝对存在任何保守数量的根本原因是,序列的一个罐子是一个线性操作员,这是一个同种程序,这是一个负责存在的rmation rmation conting rmation cantervant of sequestions cormation of ctant ctant contant operspesseriation the。运动。结构。 当前的提案着重于第一个解释,使用LAX对及其同志变形,其中一个重要的E LAX对是分布(Schwartz)Lax对。中心是局部相干模式的存在,在剖面的空间衍生物中称为pearson ular行为。由于奇异导致“ Lax-ablectable”方程的相关扩展为域办公室非平滑度,因此经典的laxalsm。 该程序的一般目标是: (1)建立N-双次字符串(n = 2,n = 3,n = 4环,一个立方字符串,欧拉 - 伯努利束)的组合变形,并可能具有广义的Cauchay Borther的作用n大于3的多项式。鉴于与地球物理反相反相反的问题相关的其他地方,Euler-Bernoulli梁具有重要意义。 (2)分类Pearmon分布宽松对和相关边界值证明; (3)构建所谓的Nikishin Systems o的主教矿石近似值O源于申请人为degasperis-procesi方程提出的立方字符串问题,并建立这些近似值与N级不均匀字符串的反向问题之间的链接。

项目成果

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Szmigielski, Jacek其他文献

Szmigielski, Jacek的其他文献

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{{ truncateString('Szmigielski, Jacek', 18)}}的其他基金

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
  • 财政年份:
    2022
  • 资助金额:
    $ 1.53万
  • 项目类别:
    Discovery Grants Program - Individual
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
  • 财政年份:
    2021
  • 资助金额:
    $ 1.53万
  • 项目类别:
    Discovery Grants Program - Individual
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
  • 财政年份:
    2019
  • 资助金额:
    $ 1.53万
  • 项目类别:
    Discovery Grants Program - Individual
Integrable nonlinear wave equations, Cauchy biorthogonal polynomials and related inverse problems
可积非线性波动方程、柯西双正交多项式及相关反问题
  • 批准号:
    RGPIN-2014-05358
  • 财政年份:
    2018
  • 资助金额:
    $ 1.53万
  • 项目类别:
    Discovery Grants Program - Individual
Integrable nonlinear wave equations, Cauchy biorthogonal polynomials and related inverse problems
可积非线性波动方程、柯西双正交多项式及相关反问题
  • 批准号:
    RGPIN-2014-05358
  • 财政年份:
    2017
  • 资助金额:
    $ 1.53万
  • 项目类别:
    Discovery Grants Program - Individual
Integrable nonlinear wave equations, Cauchy biorthogonal polynomials and related inverse problems
可积非线性波动方程、柯西双正交多项式及相关反问题
  • 批准号:
    RGPIN-2014-05358
  • 财政年份:
    2016
  • 资助金额:
    $ 1.53万
  • 项目类别:
    Discovery Grants Program - Individual
Integrable nonlinear wave equations, Cauchy biorthogonal polynomials and related inverse problems
可积非线性波动方程、柯西双正交多项式及相关反问题
  • 批准号:
    RGPIN-2014-05358
  • 财政年份:
    2015
  • 资助金额:
    $ 1.53万
  • 项目类别:
    Discovery Grants Program - Individual
Integrable nonlinear wave equations, Cauchy biorthogonal polynomials and related inverse problems
可积非线性波动方程、柯西双正交多项式及相关反问题
  • 批准号:
    RGPIN-2014-05358
  • 财政年份:
    2014
  • 资助金额:
    $ 1.53万
  • 项目类别:
    Discovery Grants Program - Individual
Integrable nonlinear equations, total positivity and biorthogonal polynomials
可积非线性方程、总正性和双正交多项式
  • 批准号:
    138591-2009
  • 财政年份:
    2013
  • 资助金额:
    $ 1.53万
  • 项目类别:
    Discovery Grants Program - Individual
Integrable nonlinear equations, total positivity and biorthogonal polynomials
可积非线性方程、总正性和双正交多项式
  • 批准号:
    138591-2009
  • 财政年份:
    2012
  • 资助金额:
    $ 1.53万
  • 项目类别:
    Discovery Grants Program - Individual

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相似海外基金

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
  • 财政年份:
    2022
  • 资助金额:
    $ 1.53万
  • 项目类别:
    Discovery Grants Program - Individual
Novel Challenges in Nonlinear Waves and Integrable Systems
非线性波和可积系统的新挑战
  • 批准号:
    2106488
  • 财政年份:
    2021
  • 资助金额:
    $ 1.53万
  • 项目类别:
    Standard Grant
Nonlinear integrable systems and representation theory -revisited-
非线性可积系统和表示论-重温-
  • 批准号:
    21K03208
  • 财政年份:
    2021
  • 资助金额:
    $ 1.53万
  • 项目类别:
    Grant-in-Aid for Scientific Research (C)
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
  • 财政年份:
    2021
  • 资助金额:
    $ 1.53万
  • 项目类别:
    Discovery Grants Program - Individual
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
  • 财政年份:
    2019
  • 资助金额:
    $ 1.53万
  • 项目类别:
    Discovery Grants Program - Individual
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