On the Korteweg‐de Vries approximation for a Boussinesq equation posed on the infinite necklace graph

Abstract

Motivated by the question how to describe long‐wave dynamics on periodic networks, we consider a Boussinesq equation posed on the infinite periodic necklace graph. For the description of long‐wave traveling waves, we derive the KdV equation and establish the validity of this formal approximation by providing estimates for the error. The proof is based on suitable energy estimates. As a consequence of the approximation result, the soliton dynamics present in the KdV equation can approximately be seen for the original system, too. There are no serious obstacles to transfer the analysis to other dispersive systems or to other periodic quantum graphs for which the KdV equation can be derived. However, a general KdV validity theory on quantum graphs would be extremely abstract and of minor use.