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

可积偏微分方程作为近似模型方程出现在流体、等离子体和光纤中波动现象的数学描述中。它们的特殊之处在于它们具有无限多的运动积分，这是丰富的数学结构的标志。在研究这一类特殊的偏微分方程的早期阶段，P. Lax指出，存在无穷多个守恒量的根本原因是，人们可以将这些方程中的每一个都与一个线性算子联系起来，该算子经历等谱（保谱）变形，导致大量运动常数的存在。这种观点最终导致了可积系统范式的形成，它将所谓的Lax对置于理论的前沿和中心。这种观点在平面连接理论和基于双哈密顿结构的辛方法中都有对应的几何观点。目前的建议集中在第一种解释上，使用Lax对及其等谱变形，并有一个重要的推广：Lax对是分布的（Schwartz） Lax对。这条研究路线的主要推动力来自于过去二十年中发现的一类令人兴奋的可积模型，这些模型表现出一些新的特征，其中心是局部相干模式的存在，称为峰子，在剖面的空间导数中具有奇异行为。peak - on方程是通过对分布Lax对的仔细考虑而得到的有趣的和意想不到的结果，即对具有奇异支持的某些类型的分布强制使用特定类型的乘法规则，从而将“Lax可积”方程扩展到经典Lax形式不能满足的显著非光滑域。本文的总体目标是：(1)建立n阶非齐次弦（n=2, n=3, n=4，分别对应于经典弦、三次弦和欧拉-伯努利梁）等谱变形的完整理论，以及n大于3时可能的广义柯西双正交多项式的作用。特别是，这将解决欧拉-伯努利光束的反问题，鉴于它与地球物理反问题的相关性，这在其他地方具有重要意义；(2)对峰分布Lax对进行分类，并给出相应的边值问题；(3)建立所谓Nikishin系统的混合Hermite-Pade近似理论，推广Nikishin系统的早期结果，这些结果起源于Degasperis-Procesi方程的申请人所表述的三次弦问题，并在这些近似与n阶非齐次弦的逆问题之间建立联系。

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.