An Overview on the Standing Waves of Nonlinear Schrödinger and Dirac Equations on Metric Graphs with Localized Nonlinearity

We summarize features and results on the problem of the existence of Ground States for the Nonlinear Schrödinger Equation on doubly-periodic metric graphs. We extend the results known for the two-dimensional square grid graph to the honeycomb, made of infinitely-many identical hexagons. Specifically, we show how the coexistence between one-dimensional and two-dimensional scales in the graph structure leads to the emergence of threshold phenomena known as dimensional crossover.

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“…p = 6 [6]). More recently, also some pioneering investigations of nonlinear Dirac equations has been initiated ( [9,10]).…”

Section: Existence Of Ground States: Resultsmentioning

confidence: 99%

“…Conversely, to get to the core of our non-existence results, let us consider inequality (10) and notice that for p = 6 it specializes to…”

Section: Existence Of Ground States In the Honeycomb: The Complete Rementioning

confidence: 99%

Quantum graphs and dimensional crossover: the honeycomb

Adami

Dovetta

Ruighi

2019

Communications in Applied and Industrial Mathematics

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“…In recent years, a large and still increasing interest has been devoted to the investigation of nonlinear dynamics on metric graphs or networks. Conceiving graphs as a meaningful model of ramified structures, and driven by physical applications, thorough investigations have been carried out first for nonlinear Schrödinger equations (NLS) (see, for instance, [1,2,29] and the review [27]), and more recently also for nonlinear Dirac equations (see [10,11]).…”

Section: Introductionmentioning

confidence: 99%

NLS ground states on metric trees: existence results and open questions

Dovetta

Serra

Tilli

2020

J. London Math. Soc.

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We consider the minimization of the NLS energy on a metric tree, either rooted or unrooted, subject to a mass constraint. With respect to the same problem on other types of metric graphs, several new features appear, such as the existence of minimizers with positive energy, and the emergence of unexpected threshold phenomena. We also study the problem with a radial symmetry constraint that is in principle different from the free problem due to the failure of the Pólya-Szegő inequality for radial rearrangements. A key role is played by a new Poincaré inequality with remainder.

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“…(i) continuity of the function (for details see (15)), (ii) "balance" of the derivatives (for details see (16)).…”

Section: Introductionmentioning

confidence: 99%

“…Following [31,41], also a simplified version of this model has recently gained a particular attention: the case of a nonlinearity localized on the compact core K of the graph (which is the subgraph consisting of all the bounded edges); namely, − u ′′ − χ K |u| p−2 u = λu (2) with Kirchhoff vertex conditions and χ K denoting the characteristic function of K. This problem has been studied in the L 2 -subcritical case in [51,52,54], while some new results on the L 2 -critical case have been presented in [24,25] (for a general overview see also [16]). Remark 1.1.…”

Section: Introductionmentioning

confidence: 99%

Nonlinear Dirac Equation on Graphs with Localized Nonlinearities: Bound States and Nonrelativistic Limit

Borrelli¹,

Carlone²,

Tentarelli³

2019

SIAM J. Math. Anal.

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In this paper we study the nonlinear Dirac (NLD) equation on noncompact metric graphs with localized Kerr nonlinearities, in the case of Kirchhoff-type conditions at the vertices. Precisely, we discuss existence and multiplicity of the bound states (arising as critical points of the NLD action functional) and we prove that, in the L 2 -subcritical case, they converge to the bound states of the NLS equation in the nonrelativistic limit. 1The attention recently attracted by the linear and the nonlinear Dirac equations is due to their applications, as effective equations, in many physical models, as in solid state physics and nonlinear optics [33,34].While originally the NLDE appeared as a field equation for relativistic interacting fermions [38], then it was used in particle physics to simulate features of quark confinement, acoustic physics, and in the context of Bose-Einstein condensates [34].Recently, it also appeared that some properties of physical models, as thin carbon structures, are well described using as an effective equation for non-relativistic electronic properties , the Dirac equation. We mention, thereupon, the seminal papers by Fefferman and Weinstein [28,29], the work of Arbunich and Sparber [10] (where a rigorous justification of linear and nonlinear equations in two-dimensional honeycomb structures is given) and the referenced therein. In addition, we

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An Overview on the Standing Waves of Nonlinear Schrödinger and Dirac Equations on Metric Graphs with Localized Nonlinearity

Cited by 17 publications

References 57 publications

Quantum graphs and dimensional crossover: the honeycomb

Quantum graphs and dimensional crossover: the honeycomb

NLS ground states on metric trees: existence results and open questions

Nonlinear Dirac Equation on Graphs with Localized Nonlinearities: Bound States and Nonrelativistic Limit

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