Airy-type evolution equations on star graphs

TL;DR

This paper defines and analyzes the Airy operator on star graphs, classifying boundary conditions for momentum-preserving and contraction semigroup generation, with implications for nonlinear wave propagation on networks.

Contribution

It introduces a rigorous definition of the Airy operator on star graphs and characterizes boundary conditions for unitary and contraction dynamics, including mass conservation constraints.

Findings

Complete classification of boundary conditions for momentum-preserving evolution.

Conditions for the Airy operator to generate a contraction semigroup.

Restrictions on boundary conditions under mass conservation.

Abstract

In the present paper the Airy operator on star graphs is defined and studied. 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., $L^2$-norm of the solution) preserving evolution on the graph. A second more general problem here solved 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.