Challenges of dispersionless integrability: Hirota type equations

无色散可积性的挑战：Hirota 型方程

基本信息

批准号：
EP/N031369/1
负责人：
Evgeny Ferapontov
金额：
$ 35.72万
依托单位：
Loughborough University
依托单位国家：
英国
项目类别：
Research Grant
财政年份：
2017
资助国家：
英国
起止时间：
2017 至无数据
项目状态：
已结题

来源：
https://gtr.ukri.org/projects?ref=EP%2FN031369%2F1
关键词：
Challenges dispersionless integrability Hirota type

项目摘要

Dispersionless systems typically arise as long-wave approximations to equations governing various physical phenomena. Applications include shallow water theory, aerodynamics, Whitham averaging theory, Laplacian growth processes, general relativity, and differential geometry. In many particularly interesting cases the resulting dispersionless systems have an additional property of integrability (informally, this means that they are amenable to analytical, not just numerical, treatment). Recently, our group has proposed a novel approach to the classification of integrable models of this kind, known as the method of hydrodynamic reductions. It is based on the requirement that the original multi-dimensional system can be decoupled into a collection of consistent 1+1 dimensional systems of hydrodynamic type in an infinity of ways. It was demonstrated that this requirement provides an efficient classification criterion. Dispersionless integrability proved to be an exciting research area with deep links to generalised conformal geometry, theory of special functions, complex analysis, algebraic geometry, and twistor theory.The key challenges of dispersionless integrability can be summarised as follows:1. Prove that the moduli spaces of dispersionless integrable systems are finite-dimensional (that is, such systems depend on finitely many essential parameters). Prove that `generic' systems of this type can be parametrised by special functions such as generalised hypergeometric functions, elliptic functions, or modular forms. 2. Prove that in 3D, every dispersionless integrable system possesses an integrable dispersive regularisation (such regularisations are known to prevent breakdown of classical solutions by generating, near the point of gradient catastrophe, a zone of rapid modulated oscillations later transforming into solitons). For `generic' dispersionless integrable systems, such regularisations constitute a novel class of fully discrete integrable equations. 3. Prove that in 4D, every dispersionless integrable system is necessarily linearly degenerate (the property of linear degeneracy is closely related to the null condition of Klainerman that insures global existence of classical solutions, even without any dispersive regularisation). 4. Develop a general solution procedure for linearly degenerate dispersionless integrable systems (non-breaking character of a linearly degenerate evolution suggests a dispersionless analogue of the classical inverse scattering transform). 5. Generalise the method of hydrodynamic reductions to systems that are not translationally invariant (the main problem here is the lack of a general theory of integrability of translationally non-invariant systems of hydrodynamic type in 1+1 dimensions). 6. Relate dispersionless integrability to generalised conformal geometry (generalised Einstein-Weyl geometry in 3D, or generalised self-dual geometry in 4D).In full generality, the problems formulated above are out of reach at present. This is primarily due to the complexity of the integrability conditions, as well as their subtle dependence on the type of system under study. In this project, we plan to address these challenges for the particularly interesting class of dispersionless Hirota type equations, which appear in applications in nonlinear acoustics (dispersionless Kadomtsev-Petviashvili equation), general relativity (Boyer-Finley equation), differential geometry (special Lagrangian submanifolds, affine hyperspheres), dispersionless limits of various integrable hierarchies of KP/Toda type, and so on. I strongly believe that successful solution of the above problems for Hirota type equations, and the relevant new analytic/geometric techniques, would significantly advance our understanding of multi-dimensional dispersionless integrability. In fact, the class of Hirota type equations is broad enough to contain all essential difficulties of general challenges.

无分散系统通常会出现与各种物理现象的方程式的长波近似。应用包括浅水理论，空气动力学，平均理论，拉普拉斯的生长过程，一般相对论和差异几何形状。在许多特别有趣的情况下，由此产生的无分散系统具有额外的集成性属性（非正式地，这意味着它们可以与分析相吻合，而不仅仅是数值，治疗）。最近，我们的小组提出了一种新的方法，以分类这种类型的综合模型，称为流体动力学降低方法。它基于以下要求，即可以将原始的多维系统分解为一致的1+1维度系统的集合，以无限的方式进行流体动力类型的系统。证明该要求提供了有效的分类标准。无分散性可集成性被证明是一个令人兴奋的研究领域，具有与广义共形几何学，特殊功能理论，复杂分析，代数几何学和扭曲器理论的密切联系。无分散整合性的关键挑战可以概括如下：1。证明无分散集成系统的模量空间是有限维度的（即，这样的系统取决于有限的许多基本参数）。证明这种类型的“通用”系统可以通过特殊功能（例如广义高几下函数，椭圆函数或模块化形式）进行参数。 2.证明，在3D中，每个无散的集成系统都具有可集成的分散正则化（已知这种正则化是通过产生梯度灾难点来防止经典溶液的崩溃，这是梯度灾难的点，后来转变为独奏子的快速调制振荡区）。对于“通用”无集成系统，此类正规化构成了一类新的完全分散的集成方程。 3.证明，在4D中，每个无散的集成系统都必须线性退化（线性退化的特性与Klainerman的无效条件密切相关，即使没有任何分散正则化，也可以确保经典解决方案的全球存在）。 4。为线性退化无集成系统（线性退化进化的非断裂特征）开发一般解决方案程序，这表明经典逆散射变换的无分散类似物）。 5。将流体动力学降低的方法概括为不是翻译不变的系统（这里的主要问题是缺乏在1+1维度中流体动力类型的翻译非不变系统的整合性的一般理论）。 6。将无分散整合性与广义的保形几何形状（3D中的广义爱因斯坦 - 韦尔几何形状，或4D中的广义自二几何形状）。在完全的一般性中，上面提出的问题目前是无法触及的。这主要是由于整合性条件的复杂性及其对所研究系统类型的微妙依赖性。 In this project, we plan to address these challenges for the particularly interesting class of dispersionless Hirota type equations, which appear in applications in nonlinear acoustics (dispersionless Kadomtsev-Petviashvili equation), general relativity (Boyer-Finley equation), differential geometry (special Lagrangian submanifolds, affine hyperspheres), dispersionless limits of various integrable KP/TODA类型的层次结构，依此类推。我坚信，对于上述问题方程的成功解决方案以及相关的新分析/几何技术，将大大提高我们对多维分散性集成性的理解。实际上，海洛塔类型方程式的类别足够广泛，可以包含一般挑战的所有基本困难。

项目成果

期刊论文数量（10）

专著数量（0）

科研奖励数量（0）

会议论文数量（0）

专利数量（0）

Integrable systems in 4D associated with sixfolds in Gr(4, 6)

与 Gr(4, 6) 中的六重相关的 4D 可积系统

DOI：
发表时间：
2018
期刊：
Int. Math. Res. Not. IMRN, DOI:10.1093/imrn/rnx308
影响因子：
0
作者：
B. Doubrov
通讯作者：
B. Doubrov

Modular Forms of Degree 2 and Curves of Genus 2 in Characteristic 2

特征 2 中的 2 次模形式和 2 格曲线

DOI：
10.1093/imrn/rnaa239
发表时间：
2022
期刊：
International Mathematics Research Notices
影响因子：
1
作者：
Cléry F
通讯作者：
Cléry F

Second-order PDEs in four dimensions with half-flat conformal structure.

具有半平坦共形结构的四维二阶偏微分方程。

DOI：
10.1098/rspa.2019.0642
发表时间：
2020
期刊：
Proceedings. Mathematical, physical, and engineering sciences
影响因子：
0
作者：
Berjawi S
通讯作者：
Berjawi S

On Christol's conjecture

关于克里斯托尔的猜想

DOI：
10.1088/1751-8121/ab82dc
发表时间：
2020
期刊：
Mathematical and Theoretical
影响因子：
0
作者：
Abdelaziz Y
通讯作者：
Abdelaziz Y

Dispersionless Hirota Equations and the Genus 3 Hyperelliptic Divisor

无色散 Hirota 方程和 Genus 3 超椭圆除数

DOI：
10.1007/s00220-019-03549-7
发表时间：
2019
期刊：
Communications in Mathematical Physics
影响因子：
2.4
作者：
Cléry F
通讯作者：
Cléry F

DOI：
{{ item.doi }}
发表时间：
{{ item.publish_year }}
期刊：
{{ item.journal_name }}
影响因子：
{{ item.factor }}
作者：
{{ item.authors }}
通讯作者：
{{ item.author }}

数据更新时间：{{ journalArticles.updateTime }}

作者：
{{ item.author }}

数据更新时间：{{ monograph.updateTime }}

作者：
{{ item.author }}

数据更新时间：{{ sciAawards.updateTime }}

作者：
{{ item.author }}

数据更新时间：{{ conferencePapers.updateTime }}

作者：
{{ item.author }}

数据更新时间：{{ patent.updateTime }}

Evgeny Ferapontov其他文献

Alexander Petrovich Veselov

亚历山大·彼得罗维奇·维谢洛夫

DOI：
发表时间：
2016
期刊：
影响因子：
0
作者：
Vsevolod Eduardovich Adler;Yu Yu Berest;V. Buchstaber;P. Grinevich;B. Dubrovin;I M Krichever;Sergey Novikov;A. Sergeev;Misha Feigin;Giovanni Felder;Evgeny Ferapontov;O. Chalykh;P. Etingof
通讯作者：
P. Etingof