Orbital stability of standing waves for supercritical NLS with potential on graphs

Ardila, Alex H.

doi:10.1080/00036811.2018.1530763

Cited by 8 publications

(10 citation statements)

References 45 publications

(90 reference statements)

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“…The δ-type conditions at one or more vertices or the presence of external potentials may give rise to a negative eigenvalue of the linear operator associated to the quadratic part of the energy. In this case, a ground state always exist in the subcritical [43], critical [40], and supercritical [25] cases. First examples of the ground state for the δ-type conditions were considered in [5,6].…”

Section: Variational Methods For the Ground Statementioning

confidence: 98%

Standing waves on quantum graphs

Kairzhan,

Noja,

Pelinovsky

2022

Preprint

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Section: Variational Methods For the Ground Statementioning

confidence: 98%

Standing waves on quantum graphs

Kairzhan,

Noja,

Pelinovsky

2022

Preprint

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“…We use the representation ω = −ε 4 and the scaling transformation (3.1) for Φ = (u, v) ∈ H 2 NK (T ). Similarly, we represent ϒ = (u, v) by using the scaling transformation 10) from which the following boundary-value problem is obtained for (U, V):…”

Section: )mentioning

confidence: 99%

“…By Theorem 2.2 in [3] for the subcritical case p ∈ (0, 2), E μ in (1.6) satisfies the bounds 10) where E R + is the energy of a half-soliton of the NLS equation on a half-line with the same mass μ and E R is the energy of a full soliton on a full line with the same mass μ. By Theorem 3.3 and Corollary 3.4 in [4], the infimum is attained if there exists…”

Section: Introductionmentioning

confidence: 98%

“…Ground states on metric graphs with Neumann-Kirchhoff boundary conditions at the vertices of G and no external potentials exist under rather restrictive topological conditions (see [3][4][5][6]). When delta-impurities at the vertices or external potentials give rise to a negative eigenvalue of the linearized operator at the zero solution, a ground state always exist in the subcritical [15], critical [13], and supercritical [10] cases.…”

Section: Introductionmentioning

confidence: 99%

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Standing waves of the quintic NLS equation on the tadpole graph

Noja

Pelinovsky

2020

Calc. Var.

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The tadpole graph consists of a circle and a half-line attached at a vertex. We analyze standing waves of the nonlinear Schrödinger equation with quintic power nonlinearity equipped with the Neumann–Kirchhoff boundary conditions at the vertex. The profile of the standing wave with the frequency $$\omega \in (-\infty ,0)$$ ω ∈ ( - ∞ , 0 ) is characterized as a global minimizer of the quadratic part of energy constrained to the unit sphere in $$L^6$$ L 6 . The set of standing waves includes the set of ground states, which are the global minimizers of the energy at constant mass ($$L^2$$ L 2 -norm), but it is actually wider. While ground states exist only for a certain interval of masses, the standing waves exist for every $$\omega \in (-\infty ,0)$$ ω ∈ ( - ∞ , 0 ) and correspond to a bigger interval of masses. It is proven that there exist critical frequencies $$\omega _1$$ ω 1 and $$\omega _0$$ ω 0 with $$-\infty< \omega _1< \omega _0 < 0$$ - ∞ < ω 1 < ω 0 < 0 such that the standing waves are the ground state for $$\omega \in [\omega _0,0)$$ ω ∈ [ ω 0 , 0 ) , local constrained minima of the energy for $$\omega \in (\omega _1,\omega _0)$$ ω ∈ ( ω 1 , ω 0 ) and saddle points of the energy at constant mass for $$\omega \in (-\infty ,\omega _1)$$ ω ∈ ( - ∞ , ω 1 ) . Proofs make use of the variational methods and the analytical theory for differential equations.

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“…On the other hand, the NLS with potential on graphs is little studied. To our knowledge, the only results concerning the existence and stability of standing waves were obtained in [5,9,10]. In the subcritical (1 < p < 5) and critical (p = 5) case orbitally stable standing waves e iωt ϕ ω (x) were constructed in [9,10] under specific conditions on V (x).…”

mentioning

confidence: 99%

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

Ardila¹,

Cely²,

Goloshchapova³

2021

Preprint

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We study the nonlinear Schrödinger equation with an arbitrary real potential V (x) ∈ (L 1 +L ∞ )(Γ) on a star graph Γ. At the vertex an interaction occurs described by the generalized Kirchhoff condition with strength −γ < 0. We show the existence of ground states ϕ ω (x) as minimizers of the action functional on the Nehari manifold under additional negativity and decay conditions on V (x). Moreover, for V (x) = − β x α , in the supercritical case, we prove that the standing waves e iωt ϕ ω (x) are orbitally unstable in H 1 (Γ) when ω is large enough. Analogous result holds for an arbitrary γ ∈ R when the standing waves have symmetric profile.

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Orbital stability of standing waves for supercritical NLS with potential on graphs

Cited by 8 publications

References 45 publications

Standing waves on quantum graphs

Standing waves on quantum graphs

Standing waves of the quintic NLS equation on the tadpole graph

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

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