Nonlinear Schrödinger equation on graphs: recent results and open problems

Abstract

In the present paper an introduction to the new subject of nonlinear dispersive hamiltonian equations on graphs is given. The focus is on recently established properties of solutions in the case of nonlinear Schrödinger equation. Special consideration is given to existence and behaviour of solitary solutions. Two subjects are discussed in some detail concerning NLS equation on a star graph: the standing waves of NLS equation on a graph with a $\delta$ interaction at the vertex; the scattering of fast solitons through an Y-junction in the cubic case. The emphasis is on description of concepts and results and on physical context, without reporting detailed proofs; some perspectives and more ambitious open problems are discussed.

20 pages; 5 figures. Added reference; corrected typos. Accepted on Phil. Trans. R. Soc. A

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Cited by in corpus (30)

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• Bound states of the NLS equation on metric graphs with localized nonlinearities
• On the lack of bound states for certain NLS equations on metric graphs
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• Soliton transport in tubular networks: transmission at vertices in the shrinking limit
• Stationary waves on nonlinear quantum graphs: General framework and canonical perturbation theory
• Standing waves on quantum graphs
• Dimensional crossover with a continuum of critical exponents for NLS on doubly periodic metric graphs
• On the Inverse Scattering Method for Integrable PDEs on a Star Graph
• Spectral stability of shifted states on star graphs
• Normalized solutions of $L^2$-supercritical NLS equations on noncompact metric graphs with localized nonlinearities
• The Korteweg-de Vries equation on a metric star graph
• Dynamics of Dirac solitons in networks
• NLS ground states on metric trees: existence results and open questions
• Nonlinear Dirac Equation On Graphs With Localized Nonlinearities: Bound States And Nonrelativistic Limit
• Airy-type evolution equations on star graphs
• Transparent nonlinear networks
• Linear instability of stationary solutions for the Korteweg-de Vries equation on a star graph
• Nonlocal Nonlinear Schrodinger Equation on Metric Graphs
• Charged solitons in branched conducting polymers
• Soliton and breather splitting on star graphs from tricrystal Josephson junctions
• Discrete Nonlocal Nonlinear Schroedinger equation on Metric Graphs: Dynamics of PT-Symmetric Solitons in Discrete Networks
• Orbital stability of standing waves for supercritical NLS with potential on graphs
• Dynamical and variational properties of the NLS-${\delta }_s^{\prime }$ equation on the star graph
• Manakov system on metric graphs: Modeling the reflectionless propagation of vector solitons in networks
• Dynamical processes on metric networks
• The Ablowitz-Ladik system on a graph
• Low-regularity well-posedness for a mixed-sign quadratic Dirac equation on $N$-star metric graphs
• Spontaneous symmetry-breaking in the nonlinear Schrödinger equation on star graphs with inhomogeneities
• Ground states for the NLS on non-compact graphs with an attractive potential