Grassmannian流及其在非线性偏微分方程中的应用
Grassmannian flows and applications to nonlinear partial differential equations
摘要：
我们展示了如何解决一个大类的偏微分方程的非局部Riccati型非线性可以产生相应的线性方程，从任意的初始数据。众所周知，演化矩阵Riccati方程可以通过将Stiefel流形上的线性演化流投影到Grassmann流形的坐标图上来生成。我们的方法依赖于将这个想法扩展到无限维的情况。关键是一个积分方程，它类似于可积系统中的马尔琴科方程，代表了坐标图。我们明确地展示了如何产生这样的解决方案，标量偏微分方程的任意阶非局部二次非线性使用我们的方法。我们提供了数值模拟，证明了非局部非线性Fisher-Kolmogorov-Petrovskii-Piskunov方程的解的产生。我们还表明，该方法可能会扩展到更一般的类的非线性偏微分系统。
英文摘要：
We show how solutions to a large class of partial differential equations with nonlocal Riccati-type nonlinearities can be generated from the corresponding linearized equations, from arbitrary initial data. It is well known that evolutionary matrix Riccati equations can be generated by projecting linear evolutionary flows on a Stiefel manifold onto a coordinate chart of the underlying Grassmann manifold. Our method relies on extending this idea to the infinite dimensional case. The key is an integral equation analogous to the Marchenko equation in integrable systems, that represents the coodinate chart map. We show explicitly how to generate such solutions to scalar partial differential equations of arbitrary order with nonlocal quadratic nonlinearities using our approach. We provide numerical simulations that demonstrate the generation of solutions to Fisher--Kolmogorov--Petrovskii--Piskunov equations with nonlocal nonlinearities. We also indicate how the method might extend to more general classes of nonlinear partial differential systems.
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