The linearized Korteweg-de Vries equation on the line with metric graph defects

We study the small amplitude linearization of the Korteweg de Vries equation on the line with a local defect scattering waves represented by a metric graph domain adjoined at one point. For a representative collection of examples, we derive explicit solution formulae expressed as contour integrals and obtain existence and unicity results for piecewise absolutely continuous data. In so doing, we implement the unified transform method on metric graphs comprising both bonds and leads for a third order differential operator.