论文标题
格拉斯曼尼亚的流量和应用于非共同非本地和局部整合系统
Grassmannian flows and applications to non-commutative non-local and local integrable systems
论文作者
论文摘要
我们提出了一种具有局部和非局部非线性的基质值非线性偏微分方程的线性化类别的方法。的确，我们概括了Pöppe最初基于求解相应的线性偏微分方程的线性化过程，以生成“散射数据”的进化Hankel操作员，然后求解类似于Marchenko方程的线性Fredholm方程，以生成Marchenko方程，以生成进化解决方案，以使进化式解决方案对非线性分化系统产生。我们的概括涉及使散射数据的基本线性部分差分系统膨胀以结合相应的伴随，反向时间或反向时空数据，并且还允许使用具有矩阵值值的内核的Hankel运算符。通过这种方法，我们将展示如何线性化矩阵非线性schrödinger和修改的Korteweg de Vries方程以及这些系统的非局部反向时间和/或反向时空版本。此外，我们制定了一个统一的线性化程序，该程序将所有这些系统作为特殊情况结合在一起。此外，我们证明了所有这样的系统是弗雷德·格拉曼尼亚（Fredholm Grassmannian）的示例。
We present a method for linearising classes of matrix-valued nonlinear partial differential equations with local and nonlocal nonlinearities. Indeed we generalise a linearisation procedure originally developed by Pöppe based on solving the corresponding underlying linear partial differential equation to generate an evolutionary Hankel operator for the `scattering data', and then solving a linear Fredholm equation akin to the Marchenko equation to generate the evolutionary solution to the nonlinear partial differential system. Our generalisation involves inflating the underlying linear partial differential system for the scattering data to incorporate corresponding adjoint, reverse time or reverse space-time data, and it also allows for Hankel operators with matrix-valued kernels. With this approach we show how to linearise the matrix nonlinear Schrödinger and modified Korteweg de Vries equations as well as nonlocal reverse time and/or reverse space-time versions of these systems. Further, we formulate a unified linearisation procedure that incorporates all these systems as special cases. Further still, we demonstrate all such systems are example Fredholm Grassmannian flows.
