Matlab file for Riccati KdV solution from Partial differential systems with non-local nonlinearities: generation and solutions

We develop a method for generating solutions to large classes of evolutionary partial differential systems with non-local nonlinearities. For arbitrary initial data, the solutions are generated from the corresponding linearized equations. The key is a Fredholm integral equation relating the linearized flow to an auxiliary linear flow. It is analogous to the Marchenko integral equation in integrable systems. We show explicitly how this can be achieved through several examples including reaction–diffusion systems with non-local quadratic nonlinearities and the nonlinear Schrödinger equation with a non-local cubic nonlinearity. In each case, we demonstrate our approach with numerical simulations. We discuss the effectiveness of our approach and how it might be extended. This article is part of the theme issue ‘Stability of nonlinear waves and patterns and related topics’.

我们开发了一种求解方法，可生成大类含非局部非线性项的演化偏微分系统的解。对于任意初值条件，其解均可通过对应的线性化方程构造得到。该方法的核心是将线性化流与辅助线性流相联系的弗雷德霍姆积分方程（Fredholm integral equation），其与可积系统中的马尔琴科积分方程（Marchenko integral equation）具有相似性。我们通过多个实例详细展示了该方法的实现路径，其中包括带非局部二次非线性项的反应扩散系统，以及带非局部三次非线性项的非线性薛定谔方程（Nonlinear Schrödinger equation）。针对每个案例，我们均通过数值模拟验证了所提方法的有效性。我们还讨论了该方法的应用效能，以及其可能的拓展方向。本文属于主题刊“非线性波与模式的稳定性及相关主题”的组成部分。