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).

korteweg de vries方程之类的可集成方程是在流体，等离子体和光纤中波浪现象的数学描述中的近似模型方程式出现的。 由于拥有无限的运动积分，它们通常伴随着深层的数学结构，因此它们是特殊的。 在过去的二十年中，已经发现了一种新的令人兴奋的模型。这些新模型表现出几种新型特征，其中心是局部，连贯模式的存在，称为pexcons，在轮廓的空间衍生物中具有奇异的行为。许多作者从许多不同的角度对许多作者进行了广泛研究Camassa-Holm（CH）方程式，其中最相关的是认识到经典时刻问题的机械提供了一个有力的工具来分析该方程式更精致的特征。在通过这种方法访问的特征中，在山峰碰撞和长时间渐近行为时，斜率陡峭。该类别的另一个方程式是大量研究兴奋的中心，是Degasperis-Procesi（DP）方程。人们意识到，除了Pearpon Solutions外，该方程还具有冲击解决方案，而且有趣的是Sharkpeakon解决方案。伴随的边界价值问题是非偏爱，导致了许多新问题和挑战。这个边界价值问题被称为立方字符串；这是三阶问题，重量是一种措施。对应于山峰的病例要求该措施离散，对于这种情况，光谱和逆光谱问题由H. lundmark和申请人解决。该问题的解决方案涉及回到T. stieltjes和M.G.的想法的结合。 Kerin对不均匀弦的研究。 对于立方字符串的逆问题，解决方案的独特特征是出现一种新型多项式的外观，鉴于Cauchy内核在生物三相关系中的存在，称为Cauchy Biorthoconal多项式。 随后澄清说，这种类型的多项式取代了正交多项式 当一个人处理某些类型的逆问题时，与非偏爱边界值问题有关。 在最后一个资金期间，由此产生的Cauchy Biorthonal多项式理论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，它旨在开发凯奇（Cauchy Biorthochonal）多项式应用的综合图表： （i）解决CH和DP的各种概括中出现的反问题； （ii）建立一种完整的机制，用于在DP方程中创建冲击峰以及Cauchy Biorthosonal多项式在从 Peakons到Shackpeakons； （iii）用cauchy内核求解随机的两个矩阵模型； （iv）了解Cauchy两个Matrix模型中的缩放定律。 另一个相关的目标，尽管不太可能直接依靠cauchy Biorthonal多项式，但是要了解B-家庭中Pearpon碰撞的性质（CH和DP方程的一个参数变形）。