The linearised Korteweg–deVries equation on general metric graphs

Abstract

We consider the linearized Korteweg-de-Vries equations, sometimes called Airy equation, on general metric graphs with edge lengths bounded away from zero. We show that properties of the induced dynamics can be obtained by studying boundary operators in the corresponding boundary space induced 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. (2010). Primary 35Q53; Secondary 47B25, 81Q35. Mathematics Subject Classification

Spectral Monotonicity for Schrödinger Operators on Metric Graphs

“…[11,41,42,43,44,45,54,62] for a non-exhaustive list, but also other types of operators such as first order operators [13,66,69], operators related to diffusion processes on metric graphs [27,28,36,67], as well as higher-order operators have been investigated as well, e.g. in [4,7,20,21,50,55,68]. Furthermore, descriptions of certain non-self-adjoint operators on metric graphs appear in the literature [26].…”