Linear instability of stationary solutions for the Korteweg-de Vries equation on a star graph

The aim of this work is to establish a linear instability criterium of stationary solutions for the Korteweg-de Vries model on a star graph with a structure represented by a finite collections of semi-infinite edges. By considering a boundary condition of $\delta$-type interaction at the graph-vertex, we show that the continuous tail and bump profiles are linearly unstable in a balanced star graph. The use of the analytic perturbation theory of operators and the extension theory of symmetric operators is a piece fundamental in our stability analysis. The arguments presented in this investigation has prospects for the study of the instability of stationary waves solutions of other nonlinear evolution equations on star graphs.