The linearized Korteweg-de-Vries equa- tion on general metric graphs

TL;DR

This paper studies the linearized Korteweg-de-Vries (Airy) equation on metric graphs, characterizing the dynamics through boundary operators and applying results to various graph structures.

Contribution

It provides a novel framework for analyzing the dynamics of the Airy equation on general metric graphs via boundary operator methods.

Findings

Characterization of unitary and contractive dynamics on graphs

Application of boundary operator approach to specific graph types

Extension of previous results to more general graph structures

Abstract

We consider the linearized Korteweg-de-Vries equa- tions, sometimes called Airy equation, on general metric graphs with edge lengths bounded away from zero. We show that pro- perties of the induced dynamics can be obtained by studying boundary operators in the corresponding boundary space indu- ced by the vertices of the graph. In particular, we characterize unitary dynamics and contractive dynamics. We demonstrate our results on various special graphs, including those recently treated in the literature.