Linearized Korteweg -- De Vries equation on a tree with unbounded root and edges

Abstract

We investigate the linearized KdV equation on a metric tree consisting of three different types of bonds: incoming unbounded root, two finite bonds, and four outgoing unbounded bonds. Under natural assumptions at the vertices, we obtain the uniqueness of a solution. To show the existence we use the theory of potentials and reduce the problem to a system of linear algebraic equations. We show that the latter is uniquely solvable under conditions of the uniqueness theorem. Also, we show that the system we consider can be used to model wave propagation in pipelines.