Non-Kirchhoff Vertices and Nonlinear Schrödinger Ground States on Graphs

Adami, Riccardo; Boni, Filippo; Ruighi, Alice

doi:10.3390/math8040617

Cited by 14 publications

(11 citation statements)

References 53 publications

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“…to [19,22,35,36,39,45] and references therein for some of the most recent developments. In the nonlinear case, a prominent focus has been devoted to Schrödinger equations (see for instance [11, 12, 21, 23, 24, 31-33, 38, 44, 53, 54, 56] as well as the recent review [3] and references therein), but other nonlinear models have been considered too (see [51] for the KdV equation and [26,27] for the Dirac equation).…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

“…The corresponding ground states problem (2) has already been addressed both on the real line [25] and on star graphs [1]. On the real line, in the regime (3) where both the nonlinearities are L 2subcritical ground states exist for every value of the mass (see [25,Theorem 1.3] and Section 2 below). The portrait is sensibly richer on star graphs.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

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Doubly nonlinear Schrödinger ground states on metric graphs

Boni¹,

Dovetta²

2021

Preprint

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We investigate the existence of ground states at prescribed mass on general metric graphs with half-lines for focusing doubly nonlinear Schrödinger equations involving both a standard power nonlinearity and delta nonlinearities located at the vertices. The problem is proved to be sensitive both to the topology and to the metric of the graph and to exhibit a phenomenology richer than in the case of the sole standard nonlinearity considered in [13,15]. On the one hand, we provide a complete topological characterization of the problem, identifying various topological features responsible for existence/non-existence of doubly nonlinear ground states in specific mass regimes. On the other hand, we describe the role of the metric in determining the exact interplay between these different topological properties.

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Section: Introduction and Main Resultsmentioning

confidence: 99%

Section: Introduction and Main Resultsmentioning

confidence: 99%

Doubly nonlinear Schrödinger ground states on metric graphs

Boni¹,

Dovetta²

2021

Preprint

Self Cite

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“…We use as test case the 2-star graph with δ and δ boundary conditions at the vertex connecting the two edges. This test case has been extensively studied from a theoretical point of view (see [24,25,32] for earlier works and [1] and the reference therein for more recent achievements). A sneak peek of the results presented in Section 5 is offered in Fig.…”

Section: • •mentioning

confidence: 99%

Gradient Flow Approach to the Calculation of Stationary States on Nonlinear Quantum Graphs

Besse,

Duboscq,

Coz

2020

Preprint

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We introduce and implement a method to compute ground states of nonlinear Schrödinger equations on metric graphs. Ground states are defined as minimizers of the nonlinear Schrödinger energy at fixed mass. Our method is based on a normalized gradient flow for the energy (i.e. a gradient flow projected on a fixed mass sphere) adapted to the context of nonlinear quantum graphs. We first prove that, at the continuous level, the normalized gradient flow is well-posed, mass-preserving, energy diminishing and converges (at least locally) toward the ground state. We then establish the link between the continuous flow and its discretezed version. We conclude by conducting a series of numerical experiments in model situations showing the good performance of the discrete flow to compute the ground state. Further experiments as well as detailled explananation of our numerical algorithm will be given in a forthcoming companion paper. Contents 1. Introduction 2. Preliminaries 2.1. Linear quantum graphs 2.2. Nonlinear quantum graphs 3. Continuous normalized gradient flow 3.1. Local well-posedness of the continuous normalized gradient flow 3.2. The normal part of the continuous normalized gradient flow 3.3. Convergence of the normal part of the continuous normalized gradient flow 4. Space-time discretization of the normalized gradient flow 4.1.

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“…As the literature on the subject witnessed a massive growth, we refrain from overviewing it here, redirecting e.g. to [1,3,[9][10][11][12][13][14][15][16]21,26,29,32,33,[38][39][40] for some of the most recent developments and to the reviews [2,34] for more comprehensive discussions. Within the whole theory, prominent efforts have been devoted to the analysis of existence of positive standing wave solutions, with a particular focus on ground states.…”

Section: Introductionmentioning

confidence: 99%

On the notion of ground state for nonlinear Schrödinger equations on metric graphs

Coster¹,

Dovetta²,

Galant³

et al. 2023

Preprint

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We compare ground states for the nonlinear Schrödinger equation on metric graphs, defined as global minimizers of the action functional constrained on the Nehari manifold, and least action solutions, namely minimizers of the action among all solutions to the equation. In principle, four alternative cases may take place: ground states do exist (thus coinciding with least action solutions); ground states do not exist while least action solutions do; both ground states and least action solutions do not exist and the levels of the two minimizing problems coincide; both ground states and least action solutions do not exist and the levels of the two minimizing problems are different. We show that in the context of metric graphs all four alternatives do occur. This is accomplished by a careful analysis of doubly constrained variational problems. As a by-product, we obtain new multiplicity results for positive solutions on a wide class of noncompact metric graphs.

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Non-Kirchhoff Vertices and Nonlinear Schrödinger Ground States on Graphs

Cited by 14 publications

References 53 publications

Doubly nonlinear Schrödinger ground states on metric graphs

Doubly nonlinear Schrödinger ground states on metric graphs

Gradient Flow Approach to the Calculation of Stationary States on Nonlinear Quantum Graphs

On the notion of ground state for nonlinear Schrödinger equations on metric graphs

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