Integrable nonlinear wave equations, Cauchy biorthogonal polynomials and related inverse problems

可积方程如Korteweg de弗里斯方程作为流体、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穷多个运动积分，通常伴随着深刻的数学结构。在过去的二十年里，人们发现了一类新的令人兴奋的这种类型的模型。这些新的模型表现出几个新的功能，其中的核心是存在的本地化，相干模式，称为峰子，奇异行为的空间导数的配置文件。第一个这样的模型方程-Camassa-Holm（CH）方程-已经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械提供了一个强大的工具来分析这个方程的更微妙的功能。其中的功能访问通过这种方法是陡峭的斜率在碰撞的峰值和长时间的渐近行为的时间。另一个方程从这类一直在中心的相当大的研究兴奋是Degasperis-Procesi（DP）方程。人们意识到，这个方程除了峰子解之外，还有激波解，非常有趣的是，还有激波峰子解。伴随的边值问题是非自伴的，从而产生了许多新的问题和挑战。这个边值问题被命名为立方弦;它是一个三阶问题的重量是一个措施。对应于峰值的情况要求该测度是离散的，对于这种情况，谱和逆谱问题由H.伦德马克和申请人。这个问题的解决方案涉及到一个可以追溯到T。Stieltjes和M.G.克莱因对非均匀弦的研究。三次弦反问题解的显著特点是出现了一种新型的多项式，由于双正交关系中存在柯西核，因此称为柯西双正交多项式。随后澄清，这种类型的多项式取代正交多项式时，一个是处理某种类型的反问题绑到非自伴边值问题。由此产生的理论柯西双正交多项式经历了相当大的发展，在过去的资助期间。拟议的研究计划是申请人过去工作的延续。部分是针对发展柯西双正交多项式应用的综合地图：（i）解决出现在CH和DP的各种推广中的反问题;（ii）建立DP方程中激波峰值产生的完整机制和柯西双正交多项式在从峰值到激波峰值过渡中的作用;（iii）求解具有柯西核的随机双矩阵模型;（iv）理解柯西双矩阵模型中的标度律。另一个相关的目标，虽然不太可能直接依赖于柯西双正交多项式，是了解峰子碰撞的性质在b族（一个参数变形的CH和DP方程）。

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).