NLS bifurcations on the bowtie combinatorial graph and the dumbbell metric graph

Goodman, Roy H.

doi:10.3934/dcds.2019093

Cited by 15 publications

(20 citation statements)

References 31 publications

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“…Turning to bound states, we point out that a detailed analysis of the stationary solutions and their stability properties has been so far carried out only for the cubic NLS on specific graphs, such as the dumbbell graph (see Figure 3, left) in [32] (we remark that the published paper contained an error that was later corrected on the arXiv version [33]). We also note the work [24] which analyzes the NLS equation on the dumbbell graph in relation with a discrete equation on the bowtie graph.…”

Section: Resultsmentioning

confidence: 99%

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

2018

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We consider the nonlinear Schrödinger equation with pure power nonlinearity on a general compact metric graph, and in particular its stationary solutions with fixed mass. Since the the graph is compact, for every value of the mass there is a constant solution. Our scope is to analyze (in dependence of the mass) the variational properties of this solution, as a critical point of the energy functional: local and global minimality, and (orbital) stability. We consider both the subcritical regime and the critical one, in which the features of the graph become relevant. We describe how the above properties change according to the topology and the metric properties of the graph.

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Section: Resultsmentioning

confidence: 99%

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

2018

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“…The analytical results also were supported by numerical computations. Later, in [14], the symmetry-preserving bifurcation described in [23] was studied in details.…”

Section: Introductionmentioning

confidence: 99%

Orbital instability of standing waves for NLS equation on star graphs

Kairzhan

2019

Proc. Amer. Math. Soc.

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We consider a nonlinear Schrödinger (NLS) equation with any positive power nonlinearity on a star graph Γ (N half-lines glued at the common vertex) with a δ interaction at the vertex. The strength of the interaction is defined by a fixed value α ∈ R. In the recent works of Adami et al., it was shown that for α = 0 the NLS equation on Γ admits the unique symmetric (with respect to permutation of edges) standing wave and that all other possible standing waves are nonsymmetric. Also, it was proved for α < 0 that, in the NLS equation with a subcritical power-type nonlinearity, the unique symmetric standing wave is orbitally stable.In this paper, we analyze stability of standing waves for both α < 0 and α > 0. By extending the Sturm theory to Schrödinger operators on the star graph, we give the explicit count of the Morse and degeneracy indices for each standing wave. For α < 0, we prove that all nonsymmetric standing waves in the NLS equation with any positive power nonlinearity are orbitally unstable. For α > 0, we prove the orbital instability of all standing waves.

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“…Additionally, we also observe a fast oscillation. We strongly suspect that it is due to the trapping mode and hence its frequency is approximately given by (16).…”

Section: Scattering Of Sg Breathersmentioning

confidence: 95%

“…The existence of ground states of the same equation on several types of star metric graphs has been considered in [13]. Bifurcations of stationary solutions in various other simple topologies have also been studied, such as in tadpole graphs consisting of a half-line joined to a loop at a single vertex [14], dumbbell-shaped metric graphs [15,16], bowtie graphs [16], and double-bridge graphs [17].…”

Section: Introductionmentioning

confidence: 99%

Soliton and Breather Splitting on Star Graphs from Tricrystal Josephson Junctions

et al. 2019

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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.

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NLS bifurcations on the bowtie combinatorial graph and the dumbbell metric graph

Cited by 15 publications

References 31 publications

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

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

Orbital instability of standing waves for NLS equation on star graphs

Soliton and Breather Splitting on Star Graphs from Tricrystal Josephson Junctions

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