Special Issue "Symmetries of Nonlinear PDEs on Metric Graphs and Branched Networks"

A special issue of Symmetry (ISSN 2073-8994).

Deadline for manuscript submissions: closed (15 January 2019).

Printed Edition Available!
A printed edition of this Special Issue is available here.

Special Issue Editors

Prof. Diego Noja
Website
Guest Editor
Dipartimento di Matematica e Applicazioni Università di Milano Bicocca Via R.Cozzi 55, 20125, Milano
Interests: singular perturbations of Schrödinger; wave and dirac equations and applications to various physical models; linear and nonlinear dispersive equations on singular structures and metric graphs; models of interaction between particles and classical fields
Prof. Dmitry Pelinovsky
Website
Guest Editor
McMaster University, Canada
Interests: partial differential equations; nonlinear wave; spectral analysis

Special Issue Information

Dear Colleagues,

This Special Issue focuses on recent progress in a new area of mathematical physics and applied analysis, namely, on nonlinear partial differential equations on metric graphs and branched networks. This subject has seen many developments in the recent years. Graphs represent a system of edges connected at one or more branching points (vertices). The connection rule determines the graph topology. When the edges can be assigned a length and the wave functions on the edges are defined in metric spaces, the graph is called a metric graph.

Evolution equations on metric graphs have attracted much attention as effective tool for the modeling of particle and wave dynamics in branched structures and networks. Because the branched structures and networks appear in different areas of contemporary physics with many applications to electronics, biology, material science and nanotechnology, developing of effective modeling tools is important for many practical problems arising in these areas.

The list of important problems includes searches for standing waves, exploring of their properties (e.g., stability and asymptotic behavior), and scattering dynamics. This Special Issue will collect a representative sample of works devoted to solutions of these and other problems.

Prof. Diego Noja
Prof. Dmitry Pelinovsky
Guest Editor

Manuscript Submission Information

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Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Symmetry is an international peer-reviewed open access monthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 1400 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Keywords

  • Symmetries of partial differential equations
  • Conserved quantities
  • Spectral analysis on metric graphs
  • Ground state
  • Existence and stability of standing waves

Published Papers (7 papers)

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Research

Open AccessArticle
Coupling Conditions for Water Waves at Forks
Symmetry 2019, 11(3), 434; https://doi.org/10.3390/sym11030434 - 24 Mar 2019
Cited by 1
Abstract
We considered the propagation of nonlinear shallow water waves in a narrow channel presenting a fork. We aimed at computing the coupling conditions for a 1D effective model, using 2D simulations and an analysis based on the conservation laws. For small amplitudes, this [...] Read more.
We considered the propagation of nonlinear shallow water waves in a narrow channel presenting a fork. We aimed at computing the coupling conditions for a 1D effective model, using 2D simulations and an analysis based on the conservation laws. For small amplitudes, this analysis justifies the well-known Stoker interface conditions, so that the coupling does not depend on the angle of the fork. We also find this in the numerical solution. Large amplitude solutions in a symmetric fork also tend to follow Stoker’s relations, due to the symmetry constraint. For non symmetric forks, 2D effects dominate so that it is necessary to understand the flow inside the fork. However, even then, conservation laws give some insight in the dynamics. Full article
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Open AccessArticle
Approximations of Metric Graphs by Thick Graphs and Their Laplacians
Symmetry 2019, 11(3), 369; https://doi.org/10.3390/sym11030369 - 12 Mar 2019
Abstract
The main purpose of this article is two-fold: first, to justify the choice of Kirchhoff vertex conditions on a metric graph as they appear naturally as a limit of Neumann Laplacians on a family of open sets shrinking to the metric graph (“thick [...] Read more.
The main purpose of this article is two-fold: first, to justify the choice of Kirchhoff vertex conditions on a metric graph as they appear naturally as a limit of Neumann Laplacians on a family of open sets shrinking to the metric graph (“thick graphs”) in a self-contained presentation; second, to show that the metric graph example is close to a physically more realistic model where the edges have a thin, but positive thickness. The tool used is a generalization of norm resolvent convergence to the case when the underlying spaces vary. Finally, we give some hints about how to extend these convergence results to some mild non-linear operators. Full article
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Open AccessArticle
Scale Invariant Effective Hamiltonians for a Graph with a Small Compact Core
Symmetry 2019, 11(3), 359; https://doi.org/10.3390/sym11030359 - 09 Mar 2019
Cited by 2
Abstract
We consider a compact metric graph of size ε and attach to it several edges (leads) of length of order one (or of infinite length). As ε goes to zero, the graph G ε obtained in this way looks like the star-graph formed [...] Read more.
We consider a compact metric graph of size ε and attach to it several edges (leads) of length of order one (or of infinite length). As ε goes to zero, the graph G ε obtained in this way looks like the star-graph formed by the leads joined in a central vertex. On G ε we define an Hamiltonian H ε , properly scaled with the parameter ε . We prove that there exists a scale invariant effective Hamiltonian on the star-graph that approximates H ε (in a suitable norm resolvent sense) as ε 0 . The effective Hamiltonian depends on the spectral properties of an auxiliary ε -independent Hamiltonian defined on the compact graph obtained by setting ε = 1 . If zero is not an eigenvalue of the auxiliary Hamiltonian, in the limit ε 0 , the leads are decoupled. Full article
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Open AccessArticle
Soliton and Breather Splitting on Star Graphs from Tricrystal Josephson Junctions
Symmetry 2019, 11(2), 271; https://doi.org/10.3390/sym11020271 - 20 Feb 2019
Cited by 1
Abstract
We consider the interactions of traveling localized wave solutions with a vertex in a star graph domain that describes multiple Josephson junctions with a common/branch point (i.e., tricrystal junctions). The system is modeled by the sine-Gordon equation. The vertex is represented by boundary [...] Read more.
We consider the interactions of traveling localized wave solutions with a vertex in a star graph domain that describes multiple Josephson junctions with a common/branch point (i.e., tricrystal junctions). The system is modeled by the sine-Gordon equation. The vertex is represented by boundary conditions that are determined by the continuity of the magnetic field and vanishing total fluxes. When one considers small-amplitude breather solutions, the system can be reduced into the nonlinear Schrödinger equation posed on a star graph. Using the equation, we show that a high-velocity incoming soliton is split into a transmitted component and a reflected one. The transmission is shown to be in good agreement with the transmission rate of plane waves in the linear Schrödinger equation on the same graph (i.e., a quantum graph). In the context of the sine-Gordon equation, small-amplitude breathers show similar qualitative behaviors, while large-amplitude ones produce complex dynamics. Full article
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Open AccessArticle
On the Nodal Structure of Nonlinear Stationary Waves on Star Graphs
Symmetry 2019, 11(2), 185; https://doi.org/10.3390/sym11020185 - 05 Feb 2019
Abstract
We consider stationary waves on nonlinear quantum star graphs, i.e., solutions to the stationary (cubic) nonlinear Schrödinger equation on a metric star graph with Kirchhoff matching conditions at the centre. We prove the existence of solutions that vanish at the centre of the [...] Read more.
We consider stationary waves on nonlinear quantum star graphs, i.e., solutions to the stationary (cubic) nonlinear Schrödinger equation on a metric star graph with Kirchhoff matching conditions at the centre. We prove the existence of solutions that vanish at the centre of the star and classify them according to the nodal structure on each edge (i.e., the number of nodal domains or nodal points that the solution has on each edge). We discuss the relevance of these solutions in more applied settings as starting points for numerical calculations of spectral curves and put our results into the wider context of nodal counting, such as the classic Sturm oscillation theorem. Full article
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Open AccessArticle
An Overview on the Standing Waves of Nonlinear Schrödinger and Dirac Equations on Metric Graphs with Localized Nonlinearity
Symmetry 2019, 11(2), 169; https://doi.org/10.3390/sym11020169 - 01 Feb 2019
Cited by 4
Abstract
We present a brief overview of the existence/nonexistence of standing waves for the NonLinear Schrödinger and the NonLinear Dirac Equations (NLSE/NLDE) on metric graphs with localized nonlinearity. First, we focus on the NLSE (both in the subcritical and the critical case) and, then, [...] Read more.
We present a brief overview of the existence/nonexistence of standing waves for the NonLinear Schrödinger and the NonLinear Dirac Equations (NLSE/NLDE) on metric graphs with localized nonlinearity. First, we focus on the NLSE (both in the subcritical and the critical case) and, then, on the NLDE highlighting similarities and differences with the NLSE. Finally, we show how the two equations are related in the nonrelativistic limit by the convergence of the bound states. Full article
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Open AccessArticle
A Note on Sign-Changing Solutions to the NLS on the Double-Bridge Graph
Symmetry 2019, 11(2), 161; https://doi.org/10.3390/sym11020161 - 01 Feb 2019
Abstract
We study standing waves of the NLS equation posed on the double-bridge graph: two semi-infinite half-lines attached at a circle. At the two vertices, Kirchhoff boundary conditions are imposed. We pursue a recent study concerning solutions nonzero on the half-lines and periodic on [...] Read more.
We study standing waves of the NLS equation posed on the double-bridge graph: two semi-infinite half-lines attached at a circle. At the two vertices, Kirchhoff boundary conditions are imposed. We pursue a recent study concerning solutions nonzero on the half-lines and periodic on the circle, by proving some existing results of sign-changing solutions non-periodic on the circle. Full article
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