arXiv subjects

Dave Smith

Publications and source records attributed to Dave Smith.

2 recordsLinked to original sources

arXiv2026

Interface problems of mixed spatial order

We solve interface problems on the line between various constant coefficient linear evolution partial differential equations. Our prototypical examples are the heat, linear Schr\"odinger, Airy, linearized Korteweg de Vries, and biharmonic Schr\"odinger equations. In each problem, one of the listed equations is posed on one spatial half line, and another on the other half line, with appropriate interface conditions. These problems are solved by means of novel extensions of Fokas's unified transform method and the explicit solution formulae are evaluated using Filon quadrature.

math.AP↗

arXiv2025

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.

math.AP↗