Vol. 11, No. 7, 2018

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ISSN: 1948-206X (e-only)
ISSN: 2157-5045 (print)
Airy-type evolution equations on star graphs

Delio Mugnolo, Diego Noja and Christian Seifert

Vol. 11 (2018), No. 7, 1625–1652
Abstract

We define and study the Airy operator on star graphs. The Airy operator is a third-order differential operator arising in different contexts, but our main concern is related to its role as the linear part of the Korteweg–de Vries equation, usually studied on a line or a half-line. The first problem treated and solved is its correct definition, with different characterizations, as a skew-adjoint operator on a star graph, a set of lines connecting at a common vertex representing, for example, a network of branching channels. A necessary condition turns out to be that the graph is balanced, i.e., there is the same number of ingoing and outgoing edges at the vertex. The simplest example is that of the line with a point interaction at the vertex. In these cases the Airy dynamics is given by a unitary or isometric (in the real case) group. In particular the analysis provides the complete classification of boundary conditions giving momentum (i.e., L2-norm of the solution) preserving evolution on the graph. A second more general problem solved here is the characterization of conditions under which the Airy operator generates a contraction semigroup. In this case unbalanced star graphs are allowed. In both unitary and contraction dynamics, restrictions on admissible boundary conditions occur if conservation of mass (i.e., integral of the solution) is further imposed. The above well-posedness results can be considered preliminary to the analysis of nonlinear wave propagation on branching structures.

Keywords
quantum graphs, Krein spaces, third-order differential operators, Airy operator, KdV equation
Mathematical Subject Classification 2010
Primary: 47B25, 81Q35, 35Q53
Milestones
Received: 31 October 2016
Revised: 11 December 2017
Accepted: 4 March 2018
Published: 20 May 2018
Authors
Delio Mugnolo
Lehrgebiet Analysis
Fakultät Mathematik und Informatik
FernUniversität in Hagen
Hagen
Germany
Diego Noja
Dipartimento di Matematica e Applicazioni
Università di Milano Bicocca
Milano
Italy
Christian Seifert
Ludwig-Maximilians-Universität München
Mathematisches Institut
München
Germany