The Korteweg-de Vries Equation on general star graphs

TL;DR

This paper proves local well-posedness for the Korteweg-de Vries equation on general star graphs, extending previous results from specific Y-junctions to more complex network structures using advanced analytical methods.

Contribution

It introduces a framework for analyzing the KdV equation on general star graphs, broadening the scope beyond previously studied specific configurations.

Findings

Established local well-posedness for KdV on general star graphs.

Developed integral formulas using forcing operator and Fourier restriction methods.

Extended prior results from Y junctions to more complex star graph structures.

Abstract

In this paper, we establish local well-posedness for the Cauchy problem associated with the Korteweg-de Vries (KdV) equation on a general metric star graph. The graph comprises m + k semi-infinite edges: k negative half-lines and m positive half-lines, all joined at a common vertex. The choice of boundary conditions is compatible with the conditions determined by the semigroup theory. The crucial point in this work is to obtain the integral formula using the forcing operator method and the Fourier restriction method of Bourgain. This work extends the results obtained by Cavalcante for the specific case of the Y junction to a more general class of star graphs.