Dynamics of solutions to the Korteweg--de Vries equation on a $\mathcal{Y}$-junction

Abstract

We study the long-time behavior of solutions to the Korteweg-de Vries equation on a Y-junction, a metric graph consisting of one incoming and two outgoing half-lines joined at a vertex. We first identify vertex conditions of separating-derivative type, together with a balance between the nonlinear coefficients and the coupling parameters, under which the problem is globally well-posed in $L^2$ for arbitrarily large data. This regime coincides with the one in which the explicit solitary waves of the junction, recently constructed by Mugnolo, Noja and Seifert (2025), exist. Using virial inequalities in the spirit of Muñoz and Ponce (2019), we then establish decay, in the sense of liminf, of the $L^2$ norm of the solution in time-dependent regions far from the solitons. In the strongly dominated linear region, which moves away from the vertex, the three edges decouple and the decay holds for every admissible choice of parameters. In the weakly dominated linear region, which stays close to the vertex, the topology of the graph produces boundary contributions at the vertex, and the decay is obtained under an algebraic condition on the coupling parameters that makes these contributions cancel. Finally, we discuss adaptations to general star graphs.