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

TL;DR

This paper studies the linearized KdV equation on a complex tree structure with unbounded and finite edges, establishing solution uniqueness and existence, and modeling wave propagation in pipelines.

Contribution

It introduces a framework for solving the linearized KdV equation on a metric tree with unbounded roots, using algebraic systems to prove existence and uniqueness.

Findings

Solution uniqueness under natural vertex conditions

Existence proven via potential theory and algebraic reduction

Modeling of wave propagation in pipeline networks

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.