Peakon integrable nonlinear equations and related approximation problems: the distributional approach.

在流体、等离子体和光纤中波动现象的数学描述中，可积偏微分方程组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，这是一个丰富的数学结构的标志。P.Lax在研究这类特殊的偏微分方程的早期阶段就指出，存在无穷多个守恒量的根本原因是，人们可以将一个线性算子与这些方程中的每个方程联系起来，该算子经历了等谱(保谱)变形，导致了一大族运动常数的存在。这一观点最终导致了可积系统范式的形成，它将所谓的松弛对放在了理论的前沿和中心。这一观点在平面连接理论中有几何上的对应，在基于双哈密顿结构存在的辛方法中也是如此。 目前的建议侧重于第一种解释，使用Lax对及其等谱变形，具有一个重要的推广：Lax对是分布(Schwartz)Lax对。这一系列研究的主要推动力来自于过去二十年中发现的一类激动人心的可积模型，这些模型表现出几个新的特征，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。Peakon方程是对分布Lax对的仔细考虑而产生的，导致了有趣和意想不到的结果，即对某些具有单一支撑的分布类强制使用特定类型的乘法规则，导致相应地将“Lax可积”方程扩展到经典Lax形式不适用的显著非光滑区域。 拟议方案的总体目标是： (1)建立了n阶非齐次弦(n=2，n=3，n=4，分别对应于经典弦、三次弦、Euler-Bernoulli梁)的等谱变形的完整理论，以及可能推广的柯西双正交多项式在n大于3时的作用。这尤其将解决Euler-Bernoulli梁的逆问题，鉴于它与地球物理逆问题的关系，它在其他地方具有重要意义； (2)对峰值分布Lax对和相关边值问题进行分类； (3)建立了所谓的Nikishin系统的混合Hermite-Pade逼近理论，推广了前人关于具有两个测度的Nikishin系统的结果，并建立了这些逼近与n阶非齐次弦的反问题之间的联系。

Integrable partial differential equations arose as approximate model equations in the mathematical description of wave phenomena in fluids, plasmas and optical fibres. They are special in that they possess infinitely many integrals of motion, a sign of a rich mathematical structure. P. Lax pointed out at the very early stage of research into this special class of partial differential equations that the underlying reason for the existence of infinitely many conserved quantities is that one can associate with each of these equations a linear operator which undergoes an isospectral (spectrum preserving) deformation responsible for the existence of a large family of constants of motion. This point of view eventually resulted in the formulation of a paradigm of integrable systems, which puts the so called Lax pairs front and centre of the theory. This point of view has its geometric counterpart in a theory of flat connections, as well as, in a symplectic approach based on the presence of bi-Hamiltonian structures. The current proposal focuses on the first interpretation, using Lax pairs and their isospectral deformations, with one significant generalization: the Lax pairs are distributional (Schwartz) Lax pairs. The main impetus for this line of research comes from a class of exciting integrable models discovered over last two decades which 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 peakon equations result from a careful consideration of distributional Lax pairs leading to interesting and unexpected results, i.e. forcing a particular type of multiplication rules for certain classes of distributions with singular support, leading to a relevant extension of “Lax integrable” equations into the domain of significant non-smoothness for which the classical Lax formalism fails. The general objectives of the proposed program are: (1) to establish a complete theory of isospectral deformations of the n-th order inhomogeneous strings (n=2, n=3, n=4, corresponding to a classical string, a cubic string, an Euler-Bernoulli beam, respectively) and the role of, possibly generalized, Cauchy biorthogonal polynomials for n greater than 3. This, in particular, would address the inverse problem for the Euler-Bernoulli beam which is of significance elsewhere in view of its relevance to the geophysical inverse problem; (2) to classify peakon distributional Lax pairs and pertinent boundary value problems; (3) to build a theory of mixed Hermite-Pade approximations of the so called Nikishin systems, generalizing earlier results on the Nikishin systems with two measures originating in the cubic string problem formulated by the applicant for the Degasperis-Procesi equation and establish a link between these approximations and inverse problems for n-order inhomogeneous strings.