The Korteweg–de Vries equation on a metric star graph

Abstract

We prove local well-posedness for the Cauchy problem associated to Kortewegde Vries equation on a metric star graph with three semi-infinite edges given by one negative half-line and two positives half-lines attached to a common vertex, for two classes of boundary conditions. The results are obtained in the low regularity setting by using the Duhamel Boundary Forcing Operator, in context of half-lines, introduced by Colliander, Kenig (2002), and extended by Holmer (2006) and Cavalcante (2017).

Stability of KdV solitons on the half-line

“…By the same time, Colliander and Kenig derived a global a priori estimate and for a non-optimal boundary condition f ∈ H 7 12 (R + ), and a conditional global well-posedness was obtained for the case s = 0. Recently, the first author [17] (using some of the Colliander-Kenig techniques) showed conditional local well-posedness for the IVP associated to the KdV equation on a simple star graph given by two positive halflines and a negative half-line attached in a common vertex.…”

“…See also [42] for other recents developments in this area. Following [17], we expect to deal with the corresponding KdV graph problem in the near future. Additionally, the so-called Kirkchoff boundary condition found in all above NLS papers, when taken in the KdV case, and to the limit case of only one half-line, naturally converges to the zero boundary condition used in this paper.…”

“…1 Figure 1. A star graph with two bonds By direct application the ideas used in [17] and [30] we believe that the following two boundary conditions…”

Boundary controllability of the Korteweg-de Vries equation on a tree-shaped network

“…It is worth to mention that the transmission conditions at the central node 0 are inspired by the recent papers [20] and [6]. It is not the only possible choice, and the main motivation is that they guarantee uniqueness of the regular solutions of the KdV equation linearized around 0 (see [6,26]).…”

“…It is worth to mention that the transmission conditions at the central node 0 are inspired by the recent papers [20] and [6]. It is not the only possible choice, and the main motivation is that they guarantee uniqueness of the regular solutions of the KdV equation linearized around 0 (see [6,26]). A characterization of boundary conditions that imply a well-posedness dynamics for the linear Airy-type evolution equation (u t = αu xxx + βu x , where α ∈ R * , β ∈ R) on star graphs of half-lines are given in [20].…”