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

在流体、等离子体和光纤中波动现象的数学描述中，可积方程(如Korteweg de Vries方程)作为近似模型方程出现。它们的特殊之处在于，它们拥有无限多的运动积分，通常伴随着深刻的数学结构。*在过去的二十年里，发现了一类新的这种类型的令人兴奋的型号。这些新模型展示了几个新的特征，其中心是局域相干模的存在，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-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，其中最重要的是认识到经典矩问题的机制为分析该方程的更精细的特征提供了强有力的工具。这种方法可以获得的特征包括峰子碰撞时的坡度变陡和长时间的渐近行为。这类方程中的另一个一直处于相当大的研究兴奋中心的是DeGasperis-Procesi(DP)方程。人们意识到，这个方程除了峰解外，还有激波解，非常有趣的是，还有激波峰解。伴随而来的边值问题是非自伴的，导致了一系列新的问题和挑战。这个边值问题被称为三次弦；它是一个以权为度量的三阶问题。与峰值相对应的情况要求这种测量是离散的，对于这种情况，谱和逆谱问题由H.Lundmark和申请人解决。这个问题的解决涉及到T.Stieltjes和M.G.Krein对非均匀弦的研究的想法的结合。由于双正交关系中柯西核的存在，三次弦反问题的解的显著特点是出现了一类新的多项式，称为柯西双正交多项式。后来已阐明，当处理与非自伴边值问题有关的某些类型的逆问题时，这种类型的多项式取代了正交多项式。由此产生的柯西双正交多项式理论在上一个资助期有了长足的发展。**拟议的研究计划是申请者过去工作的延续。在一定程度上，它旨在开发柯西双正交多项式在以下方面的全面应用图：*(I)解决在CH和DP的各种推广中出现的反问题；*(Ii)建立在DP方程中产生激波尖峰的完整机制，以及柯西双正交多项式在从*尖峰到激峰的转变中的作用；*(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，尽管不太可能直接依赖于柯西双正交多项式，是为了了解b-族中峰子碰撞的性质(CH和DP方程的单参数变形)。

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).