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
• 负责人: Szmigielski, Jacek
• 金额: $ 1.31万
• 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: 2015
• 资助国家: 加拿大
• 起止时间: 2015-01-01 至 2016-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=588363
• 关键词: Integrable; nonlinear; wave; equations; Cauchy

Integrable equations such the Korteweg de Vries equation arise as approximate model equations in the mathematical description of wave phenomena in fluids, plasmas and optical fibres. They are special by virtue of possessing infinitely many integrals of motion, usually accompanied by deep mathematical structure. In the last two decades a new class of exciting models of this type has been discovered. These new models 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 first such model equation-the Camassa-Holm (CH) equation-has been studied extensively by many authors from many different points of view, the most pertinent of which is a realization that the machinery of the classical moment problem provides a powerful tool to analyze more delicate features of this equation. Among the features accessible by this method are steepening of the slope at the time of the collision of peakons and the long time asymptotic behaviour. Another equation from this class which has been in the center of considerable research excitement is the Degasperis-Procesi (DP) equation. It was realized that this equation has, in addition to peakon solutions, also the shock solutions and, very interestingly, the shockpeakon solutions. The accompanying boundary value problem is non-selfadjoint, resulting in a host of new problems and challenges. This boundary value problem was named the cubic string; it is a third order problem with a weight being a measure. The case corresponding to peakons requires that this measure be discrete and for such a case the spectral and the inverse spectral problem were solved by H. Lundmark and the applicant. The solution of the problem involved a combination of ideas going back to T. Stieltjes and to M.G. Krein's study of an inhomogeneous string. The distinct feature of the solution to the inverse problem for the cubic string is the appearance of a new type of polynomials, named Cauchy biorthogonal polynomials in view of the presence of the Cauchy kernel in the biorthogonality relation. It has subsequently been clarified that this type of polynomials replaces orthogonal polynomials when one is dealing with certain type of inverse problems tied to non-selfadjoint boundary value problems. The resulting theory of Cauchy biorthogonal polynomials has undergone a considerable development in the last funding period. The proposed research program is a continuation of the applicant's past work. In part, it is directed at developing a comprehensive map of applications of Cauchy biorthogonal polynomials to: (i) solving inverse problems appearing in a variety of generalizations of CH and DP; (ii) establishing a complete mechanism for the creation of shock peakons in the DP equation and the role of Cauchy biorthogonal polynomials in the transition from peakons to shockpeakons; (iii) solving random two-matrix models with Cauchy kernel; (iv) understanding the scaling laws in Cauchy two-matrix models. Another related objective, although not likely to rely directly on Cauchy biorthogonal polynomials, is to understand the nature of peakon collisions in the b-family (a one parameter deformation of the CH and DP equations).

可积方程（例如 Korteweg de Vries 方程）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出现。 它们的特殊之处在于拥有无限多个运动积分，通常伴随着深刻的数学结构。 在过去的二十年里，人们发现了一类令人兴奋的新型模型。这些新模型表现出几个新颖的特征，其核心是存在局部相干模式，称为峰子，在轮廓的空间导数中具有奇异行为。第一个这样的模型方程——卡马萨-霍尔姆 (CH) 方程——已被许多作者从许多不同的角度进行了广泛的研究，其中最相关的是认识到经典矩问题的机制提供了一个强大的工具分析这个方程的更微妙的特征。通过这种方法可以获得的特征包括峰子碰撞时斜率的陡峭和长时间渐近行为。此类中的另一个方程是 Degasperis-Procesi (DP) 方程，它一直是令人兴奋的研究中心。人们意识到，这个方程除了peakon解之外，还有冲击解，非常有趣的是，还有shockpeakon解。伴随的边值问题是非自伴的，导致了许多新的问题和挑战。这个边值问题被命名为三次弦；这是一个以权重为度量的三阶问题。对应于peakon的情况要求该测量是离散的，并且对于这种情况，谱和逆谱问题由H.Lundmark和申请人解决。该问题的解决涉及 T. Stieltjes 和 M.G. 的想法的结合。克赖因对非均匀弦的研究。 三次弦反问题解的显着特征是出现了一种新型多项式，鉴于双正交关系中柯西核的存在，称为柯西双正交多项式。 随后澄清了这种类型的多项式取代了正交多项式 当人们处理与非自伴边值​​问题相关的某种类型的反问题时。 由此产生的柯西双正交多项式理论在上一个资助期间经历了相当大的发展。 拟议的研究计划是申请人过去工作的延续。 在某种程度上，它旨在开发柯西双正交多项式应用的综合图： (i) 解决CH和DP的各种推广中出现的逆问题； (ii) 建立了在 DP 方程中创建激波峰值的完整机制以及柯西双正交多项式在从 Peakons 到 Shockpeakons； (iii) 用柯西核求解随机二矩阵模型； (iv) 了解柯西二矩阵模型中的标度定律。 另一个相关目标，虽然不太可能直接依赖柯西双正交多项式，但是了解 b 族中 Peakon 碰撞的性质（CH 和 DP 方程的单参数变形）。