Variational and Stability Properties of Constant Solutions to the NLS Equation on Compact Metric Graphs

Cacciapuoti, Claudio; Dovetta, Simone; Serra, Enrico

doi:10.1007/s00032-018-0288-y

Cited by 36 publications

(41 citation statements)

References 39 publications

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“…), but since such a dependence is not crucial we omit it for the sake of simplicity. In addition, in the following, we will always make the assumptions (18), (19) and (20) on the parameters (c n ), (ω n ). In particular, those assumptions immediately imply that…”

Section: Nonrelativistic Limit Of Solutionsmentioning

confidence: 99%

“…Lemma 4.1. Under the assumptions (18), (19) and (20), the sequence (ψ n ) is bounded in L p (K, C 2 ) (uniformly with respect to n), as well as the associated minimax levels (α n N ). Proof.…”

Section: 1mentioning

confidence: 99%

“…We also mention some interesting results on the problem of the bound states on compact graphs. For a purely variational approach we recall, e.g., [23,19], whereas for a bifurcation approach we recall, e.g., [40].…”

Section: Introductionmentioning

confidence: 99%

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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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Section: Nonrelativistic Limit Of Solutionsmentioning

confidence: 99%

Section: 1mentioning

confidence: 99%

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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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“…For the sake of completeness, we also remark that the NLSE on compact graphs (which, in particular, do not fulfill (H1)) has been studied, e.g., in [20,24,28,38]; while the case of one or higher-dimensional periodic graphs (which, in particular, do not fulfill (H2) as, for instance, in Figure 3) has been addressed, e.g., by [6,25,32,43].…”

Section: Introductionmentioning

confidence: 99%

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

2019

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We present a brief overview on the existence/nonexistence of standing waves for the NonLinear Schrödinger and the NonLinear Dirac Equations (NLSE/NLDE) on metric graphs with localized nonlinearity. We first 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, proving the convergence of bound states.

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“…The present work is devoted to stability of standing waves in the NLS equation defined on a metric graph, a subject that has seen many recent developments [18]. Existence and variational characterization of standing waves was developed for star graphs [1,2,3,4] and for general Date: February 12, 2019. metric graphs [5,6,7,8,9]. Bifurcations and stability of standing waves were further explored for tadpole graphs [19], dumbbell graphs [12,16], double-bridge graphs [20], and periodic ring graphs [10,11,21,22].…”

Section: Introductionmentioning

confidence: 99%

Drift of Spectrally Stable Shifted States on Star Graphs

Kairzhan¹,

Pelinovsky²,

Goodman³

2019

SIAM J. Appl. Dyn. Syst.

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When the coefficients of the cubic terms match the coefficients in the boundary conditions at a vertex of a star graph and satisfy a certain constraint, the nonlinear Schrödinger (NLS) equation on the star graph can be transformed to the NLS equation on a real line. Such balanced star graphs have appeared in the context of reflectionless transmission of solitary waves. Steady states on such balanced star graphs can be translated along the edges with a translational parameter and are referred to as the shifted states. When the star graph has exactly one incoming edge and several outgoing edges, the steady states are spectrally stable if their monotonic tails are located on the outgoing edges. These spectrally stable states are degenerate minimizers of the action functional with the degeneracy due to the translational symmetry. Nonlinear stability of these spectrally stable states has been an open problem up to now. In this paper, we prove that these spectrally stable states are nonlinearly unstable because of the irreversible drift along the incoming edge towards the vertex of the star graph. When the shifted states reach the vertex as a result of the drift, they become saddle points of the action functional, in which case the nonlinear instability leads to their destruction. In addition to rigorous mathematical results, we use numerical simulations to illustrate the drift instability and destruction of the shifted states on the balanced star graph.

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Variational and Stability Properties of Constant Solutions to the NLS Equation on Compact Metric Graphs

Cited by 36 publications

References 39 publications

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

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

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

Drift of Spectrally Stable Shifted States on Star Graphs

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