Instability of ground states for the NLS equation with potential on the star graph

Ardila, Alex H.; Cely, Liliana; Goloshchapova, Nataliia

doi:10.1007/s00028-021-00670-w

Cited by 7 publications

(3 citation statements)

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“…The δ-type conditions at one or more vertices or the presence of external potentials V may give rise to a negative eigenvalue of the linear operator associated with the quadratic part of the energy. In this case, a ground state always exist in the subcritical [45], critical [42], and supercritical [25,26] cases. First examples of the ground state for the δ-type conditions were considered in [5,6].…”

Section: δ-Type Conditions and External Potentialsmentioning

confidence: 98%

Standing waves on quantum graphs

Kairzhan

Noja

Pelinovsky

2022

J. Phys. A: Math. Theor.

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Section: δ-Type Conditions and External Potentialsmentioning

confidence: 98%

Standing waves on quantum graphs

Kairzhan

Noja

Pelinovsky

2022

J. Phys. A: Math. Theor.

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“…Remark 3.5. In [11] the NLS equation with the δ coupling and decaying potential V (x) on Γ i∂ t u(t) = −∆ γ u(t) + V (x)u(t) − |u(t)| p−1 u(t) has been considered. Here γ < 0 and…”

Section: Existence Of the Ground Statementioning

confidence: 99%

Dynamical and variational properties of the NLS-$δ'_s$ equation on the star graph

Goloshchapova

2022

Preprint

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We study the nonlinear Schrödinger equation with δ ′ s coupling of intensity β ∈ R \ {0} on the star graph Γ consisting of N half-lines. The nonlinearity has the form g(u) = |u| p−1 u, p > 1. In the first part of the paper, under certain restriction on β, we prove the existence of the ground state solution as a minimizer of the action functional S ω on the Nehari manifold. It appears that the family of critical points which contains a ground state consists of N profiles (one symmetric and N −1 asymmetric). In particular, for the attractive δ ′ s coupling (β < 0) and the frequency ω above a certain threshold, we managed to specify the ground state.The second part is devoted to the study of orbital instability of the critical points. We prove spectral instability of the critical points using Grillakis/Jones Instability Theorem. Then orbital instability for p > 2 follows from the fact that data-solution mapping associated with the equation is of class C 2 in H 1 (Γ). Moreover, for p > 5 we complete and concertize instability results showing strong instability (by blow up in finite time) for the particular critical points.

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“…The effect of the δ(x)-potential on the dynamics of the nonlinear Schrödinger equation has been studied intensively in later years. The Cauchy problem, existence of ground states and their stability/instability, long time dynamics (scattering and global existence, blow-up) to (1.1) with data below the ground state threshold, etc., have been studied in recent years; see for example [2,3,11,12,14,16] for more details.…”

Section: Introductionmentioning

confidence: 99%