IST versus PDE, a comparative study

Benjamin-Ono and Intermediate Long Wave equation : modeling, IST and PDE

This survey is focused on two asymptotic models for internal waves, the Benjamin-Ono (BO) and Intermediate Long Wave (ILW) equations that are integrable by inverse scattering techniques (IST). After recalling briefly their derivations we will review old and recent results on the Cauchy problem, comparing those obtains by IST and PDE techniques and also results more connected to the physical origin of the equations. We will consider mainly the Cauchy problem on the real line with only a few comments on the perodic case. We will also briefly discuss some relevant problems, in particular the high…

通过求积分求解具有亚纯势的Zakharov-Shabat系统的可解性

Solvability of the Zakharov-Shabat systems with meromorphic potentials by quadrature

本文研究了具有亚纯势的二维Zakharov-Shabat系统的可解性，证明了在势函数绝对可积条件下，该系统可通过求积法求解当且仅当势函数无反射。

采用物理信息神经网络求解Newell-Whitehead-Segel方程与Allen-Cahn方程：与样条方法的对比分析

Solving Newell-Whitehead-Segel and Allen-Cahn Equations Employing Physics-Informed Neural Networks: A Comparative Analysis with Spline Methods

本研究针对NWS与Allen-Cahn两类基础偏微分方程，将PINN方法与最优样条数值解法开展全面对比，验证了PINN求解表现更优且具备计算效率优势。

利用神经网络从数据中学习偏微分方程的功能组件

Learning functional components of PDEs from data using neural networks

本文提出利用神经网络从数据中学习偏微分方程的功能组件，通过非局部聚合-扩散方程案例研究，展示如何从稳态数据中恢复未知函数。