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

TL;DR

This paper establishes a linear instability criterion for stationary solutions of the Korteweg-de Vries equation on star graphs, showing that certain profiles are unstable under specific boundary conditions using operator theory.

Contribution

It introduces a new instability criterion for KdV solutions on star graphs with delta-type boundary conditions, employing analytic perturbation and extension theory of operators.

Findings

Continuous tail and bump profiles are linearly unstable in a balanced star graph.

The methods used have potential applications to other nonlinear evolution equations on star graphs.

The study advances understanding of stability properties of solutions on network-like structures.

Abstract

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.