## Analysis & PDE

• Anal. PDE
• Volume 11, Number 7 (2018), 1625-1652.

### Airy-type evolution equations on star graphs

#### 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., $L 2$-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.

#### Article information

Source
Anal. PDE, Volume 11, Number 7 (2018), 1625-1652.

Dates
Revised: 11 December 2017
Accepted: 4 March 2018
First available in Project Euclid: 15 January 2019

https://projecteuclid.org/euclid.apde/1547521407

Digital Object Identifier
doi:10.2140/apde.2018.11.1625

Mathematical Reviews number (MathSciNet)
MR3810468

Zentralblatt MATH identifier
06881629

#### Citation

Mugnolo, Delio; Noja, Diego; Seifert, Christian. Airy-type evolution equations on star graphs. Anal. PDE 11 (2018), no. 7, 1625--1652. doi:10.2140/apde.2018.11.1625. https://projecteuclid.org/euclid.apde/1547521407

#### References

• K. Ammari and E. Crépeau, “Feedback stabilization and boundary controllability of the Korteweg–de Vries equation on a star-shaped network”, preprint, 2017.
• W. Arendt, R. Chill, C. Seifert, J. Voigt, and H. Vogt, “Form methods for evolution equations”, lecture notes, 2015, https://www.mat.tuhh.de/veranstaltungen/isem18/pdf/LectureNotes.pdf.
• G. Berkolaiko and P. Kuchment, Introduction to quantum graphs, Mathematical Surveys and Monographs 186, Amer. Math. Soc., Providence, RI, 2013.
• J. L. Bona and R. C. Cascaval, “Nonlinear dispersive waves on trees”, Can. Appl. Math. Q. 16:1 (2008), 1–18.
• J. L. Bona, S. M. Sun, and B.-Y. Zhang, “A non-homogeneous boundary-value problem for the Korteweg–de Vries equation in a quarter plane”, Trans. Amer. Math. Soc. 354:2 (2002), 427–490.
• J. L. Bona, S. M. Sun, and B.-Y. Zhang, “A nonhomogeneous boundary-value problem for the Korteweg–de Vries equation posed on a finite domain”, Comm. Partial Differential Equations 28:7-8 (2003), 1391–1436.
• C. Cacciapuoti, D. Finco, and D. Noja, “Ground state and orbital stability for the NLS equation on a general starlike graph with potentials”, Nonlinearity 30:8 (2017), 3271–3303.
• R. Carlson, “Inverse eigenvalue problems on directed graphs”, Trans. Amer. Math. Soc. 351:10 (1999), 4069–4088.
• M. Cavalcante, “The Korteweg–de Vries equation on a metric star graph”, preprint, 2017.
• T. Colin and J.-M. Ghidaglia, “An initial-boundary value problem for the Korteweg–de Vries equation posed on a finite interval”, Adv. Differential Equations 6:12 (2001), 1463–1492.
• J. E. Colliander and C. E. Kenig, “The generalized Korteweg–de Vries equation on the half line”, Comm. Partial Differential Equations 27:11-12 (2002), 2187–2266.
• W. Craig and J. Goodman, “Linear dispersive equations of Airy type”, J. Differential Equations 87:1 (1990), 38–61.
• B. Deconinck, N. E. Sheils, and D. A. Smith, “The linear KdV equation with an interface”, Comm. Math. Phys. 347:2 (2016), 489–509.
• M. A. Dritschel and J. Rovnyak, “Extension theorems for contraction operators on Kreĭ n spaces”, pp. 221–305 in Extension and interpolation of linear operators and matrix functions, edited by I. Gohberg, Oper. Theory Adv. Appl. 47, Birkhäuser, Basel, 1990.
• K.-J. Engel and R. Nagel, One-parameter semigroups for linear evolution equations, Graduate Texts in Mathematics 194, Springer, 2000.
• P. Exner, “Momentum operators on graphs”, pp. 105–118 in Spectral analysis, differential equations and mathematical physics: a festschrift in honor of Fritz Gesztesy's 60th birthday, edited by H. Holden et al., Proc. Sympos. Pure Math. 87, Amer. Math. Soc., Providence, RI, 2013.
• A. V. Faminskii, “Quasilinear evolution equations of the third order”, Bol. Soc. Parana. Mat. $(3)$ 25:1-2 (2007), 91–108.
• A. S. Fokas, A unified approach to boundary value problems, CBMS-NSF Regional Conference Series in Applied Mathematics 78, SIAM, Philadelphia, PA, 2008.
• A. S. Fokas, A. A. Himonas, and D. Mantzavinos, “The Korteweg–de Vries equation on the half-line”, Nonlinearity 29:2 (2016), 489–527.
• N. Hayashi and E. Kaikina, Nonlinear theory of pseudodifferential equations on a half-line, North-Holland Mathematics Studies 194, Elsevier Science, Amsterdam, 2004.
• E. Hille and R. S. Phillips, Functional analysis and semi-groups, American Mathematical Society Colloquium Publications 31, Amer. Math. Soc., Providence, RI, 1957.
• J. Holmer, “The initial-boundary value problem for the Korteweg–de Vries equation”, Comm. Partial Differential Equations 31:7-9 (2006), 1151–1190.
• P. M. Jacovkis, “One-dimensional hydrodynamic flow in complex networks and some generalizations”, SIAM J. Appl. Math. 51:4 (1991), 948–966.
• D. J. Korteweg and G. de Vries, “On the change of form of long waves advancing in a rectangular canal, and on a new type of long stationary waves”, Philos. Mag. $(5)$ 39:240 (1895), 422–443.
• D. Lannes, The water waves problem, Mathematical Surveys and Monographs 188, Amer. Math. Soc., Providence, RI, 2013. Mathematical analysis and asymptotics.
• F. Linares and G. Ponce, Introduction to nonlinear dispersive equations, Springer, 2009.
• J. R. Miller, “Spectral properties and time decay for an Airy operator with potential”, J. Differential Equations 141:1 (1997), 102–121.
• D. Mugnolo, Semigroup methods for evolution equations on networks, Springer, 2014.
• D. Mugnolo and J.-F. Rault, “Construction of exact travelling waves for the Benjamin–Bona–Mahony equation on networks”, Bull. Belg. Math. Soc. Simon Stevin 21:3 (2014), 415–436.
• A. Nachbin and V. da Silva Simões, “Solitary waves in open channels with abrupt turns and branching points”, J. Nonlinear Math. Phys. 19:suppl. 1 (2012), art. id. 1240011.
• A. Nachbin and V. S. Simões, “Solitary waves in forked channel regions”, J. Fluid Mech. 777 (2015), 544–568.
• D. Noja, “Nonlinear Schrödinger equation on graphs: recent results and open problems”, Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 372:2007 (2014), art. id. 20130002.
• K. Schmüdgen, Unbounded self-adjoint operators on Hilbert space, Graduate Texts in Mathematics 265, Springer, 2012.
• C. Schubert, C. Seifert, J. Voigt, and M. Waurick, “Boundary systems and (skew-)self-adjoint operators on infinite metric graphs”, Math. Nachr. 288:14-15 (2015), 1776–1785.
• Z. A. Sobirov, M. I. Akhmedov, O. V. Karpova, and B. Jabbarova, “Linearized KdV equation on a metric graph”, Nanosystems 6:6 (2015), 757–761.
• Z. A. Sobirov, M. I. Akhmedov, and H. Uecker, “Cauchy problem for the linearized KdV equation on general metric star graphs”, Nanosystems 6:2 (2015), 198–204.
• Z. A. Sobirov, H. Uecker, and M. I. Akhmedov, “Exact solutions of the Cauchy problem for the linearized KdV equation on metric star graphs”, Uzbek. Mat. Zh. 2015:3 (2015), 143–154.
• J. J. Stoker, Water waves: the mathematical theory with applications, Pure and Applied Mathematics 4, Interscience, New York, 1957.
• G. G. Stokes, “On the theory of oscillatory waves”, Trans. of Cambridge Phil. Soc. 8:4 (1847), 441–473.