A Nonhomogeneous Boundary-Value Problem For The Nonlinear KdV Equation on Star Graphs
Abstract
This paper investigates a boundary-value problem for the Korteweg-de Vries (KdV) equation on a star-graph structure. We develop a unified framework introducing the notion of -compatibility, which generalizes classical compatibility conditions to star-shaped and more complex graph configurations, inspired by the works of Bona, Sun, and Zhang [14]. By combining analytical techniques with a fixed-point argument, we establish sharp global well-posedness for both the linear and nonlinear problems at the level. In this setting, our results extend the classical analysis for a single KdV equation [14] to star-shaped graphs composed of equations. These results provide the first comprehensive well-posedness theory for KdV equations with coupled boundary conditions on graphs. Although control issues are not treated in this article, the analytic results obtained here address several open problems, which will be addressed in a forthcoming
Cite
@article{arxiv.2512.15307,
title = {A Nonhomogeneous Boundary-Value Problem For The Nonlinear KdV Equation on Star Graphs},
author = {Roberto de A. Capistrano Filho and Hugo Parada and Jandeilson Santos da Silva},
journal= {arXiv preprint arXiv:2512.15307},
year = {2025}
}
Comments
40 pages. Comments are welcome