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.

我们提出了一种用于线性化具有局部和非局部非线性的矩阵值非线性偏微分方程类的方法。实际上，我们对Pöppe最初开发的线性化过程进行了通用化，其基础是求解相应的基础线性偏微分方程以生成“散射数据”的演化汉克算子，然后求解类似于Marchenko方程的线性Fredholm方程以生成演化解到非线性偏微分系统。我们的归纳涉及为散射数据增加基本的线性偏微分系统，以合并相应的伴随，逆时或逆时空数据，并且还允许Hankel算子具有矩阵值的核。通过这种方法，我们展示了如何线性化矩阵非线性Schrödinger和修正的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，我们制定了统一的线性化程序，将所有这些系统合并为特例。更进一步，我们证明了所有此类系统都是Fredholm Grassmannian流的示例。