Linear instability criterion for the Korteweg–de Vries equation on metric star graphs

Abstract

The aim of this work is to establish a novel linear instability criterion for the Korteweg–de Vries (KdV) model on metric graphs. In the case of balanced graphs with a structure represented by a finite collection of semi-infinite edges and with boundary condition of δ-type interaction at the graph-vertex, we show that the continuous tail and bump profiles are linearly unstable. In this case, the use of the analytic perturbation theory of operators as well as the extension theory of symmetric operators is fundamental in our stability analysis. The arguments showed in this investigation have prospects in the study of the instability of stationary waves solutions for nonlinear evolution equations on metric graph.