Airy-type evolution equations on star graphs

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.

References in corpus (1)

• Ground state and orbital stability for the NLS equation on a general starlike graph with potentials

Cited by in corpus (13)

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• Ground states of the $L^2$-critical NLS equation with localized nonlinearity on a tadpole graph
• Linear instability of stationary solutions for the Korteweg-de Vries equation on a star graph
• Lower regularity solutions of the biharmonic Schrödinger equation in a quarter plane
• An overview on the standing waves of nonlinear Schrödinger and Dirac equations on metric graphs with localized nonlinearity
• Unstable kink-soliton profiles for the sine-Gordon equation on a $\mathcal{Y}$-junction graph with $\delta$-interaction
• Forcing operators on star graphs applied for the cubic fourth order Schrödinger equation
• Controllability for Schrödinger type system with mixed dispersion on compact star graphs
• Normalized NLS ground states on a double plane hybrid
• The Korteweg-de Vries Equation on general star graphs