Standing waves on a flower graph

Kairzhan, Adilbek; Marangell, Robert; Pelinovsky, Dmitry E.; Xiao, Ke Liang

doi:10.1016/j.jde.2020.09.010

Cited by 22 publications

(54 citation statements)

References 13 publications

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“…Theorem 1 states existence of the N -pulse edge-localized states for every 1 ≤ N ≤ L and Theorem 2 states that Morse index of this N -pulse state is N . This coincides with the main results of [17] (Theorems 1, 2, and 3) in the limit ω → −∞ obtained with long analysis of the period function in the case of loops of the same normalized length.…”

Section: Examplessupporting

confidence: 91%

“…Every z ∈ (0, T + ( p, q)) can be represented by z = T + (P, Q) for some point (P, Q) ∈ E β with P ∈ ( p, p + ). By Lemma 3.8 in [17], we have s(z) = 0 if and only if ∂ Q T + (P, Q) = 0, which also follows from the first equation in system (2.20). Then, by Lemmas 3.6, 3.7, 3.9, 3.10 in [17], for sufficiently small p and q there is exactly one point (P, Q) ∈ E β with P ∈ ( p, p + ), where s(z) = s(T + (P, Q)) = 0.…”

Section: Lemma 4 Let U Be a Real Positive And Decreasing Solution To The Nonlinear Equationmentioning

confidence: 71%

“…This limit for large μ = Q( ) can be recast as the limit of large negative ω in the stationary NLS equation (1.5) for the cubic (L 2 -subcritical) nonlinearity. Existence of such edge-localized states was confirmed analytically and/or numerically for the tadpole graph [2,8,9,23,25], the dumbbell graph [14,21], and the flower graph [17]. The importance of the positive edge-localized states is motivated by their (possible) orbital stability in the NLS time flow.…”

Section: Introductionmentioning

confidence: 79%

“…The second property in (2.13) is also proven in Lemma 3.6 of [17] by using more complicated analysis of the integrals in (2.12).…”

Section: Remark 12mentioning

confidence: 86%

“…The paper is organized as follows. Section 2 reviews preliminary results from [6] and [17]. Sections 3 and 4 give the proof of Theorems 1 and 2 respectively.…”

Section: Remarkmentioning

confidence: 99%

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Multi-pulse edge-localized states on quantum graphs

Kairzhan

Pelinovsky

2021

Anal.Math.Phys.

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We construct the edge-localized stationary states of the nonlinear Schrödinger equation on a general quantum graph in the limit of large mass. Compared to the previous works, we include arbitrary multi-pulse positive states which approach asymptotically a composition of N solitons, each sitting on a bounded (pendant, looping, or internal) edge. We give sufficient conditions on the edge lengths of the graph under which such states exist in the limit of large mass. In addition, we compute the precise Morse index (the number of negative eigenvalues in the corresponding linearized operator) for these multi-pulse states. If N solitons of the edge-localized state reside on the pendant and looping edges, we prove that the Morse index is exactly N . The technical novelty of this work is achieved by avoiding elliptic functions (and related exponentially small scalings) and closing the existence arguments in terms of the Dirichlet-to-Neumann maps for relevant parts of the given graph. We illustrate the general results with three examples of the flower, dumbbell, and single-interval graphs.

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Section: Examplessupporting

confidence: 91%

Section: Lemma 4 Let U Be a Real Positive And Decreasing Solution To The Nonlinear Equationmentioning

confidence: 71%

Section: Introductionmentioning

confidence: 79%

“…The second property in (2.13) is also proven in Lemma 3.6 of [17] by using more complicated analysis of the integrals in (2.12).…”

Section: Remark 12mentioning

confidence: 86%

“…The paper is organized as follows. Section 2 reviews preliminary results from [6] and [17]. Sections 3 and 4 give the proof of Theorems 1 and 2 respectively.…”

Section: Remarkmentioning

confidence: 99%

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Multi-pulse edge-localized states on quantum graphs

Kairzhan

Pelinovsky

2021

Anal.Math.Phys.

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Stability theory for the NLS equation on looping edge graphs

Angulo Pava

2024

Math. Z.

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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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Standing waves on a flower graph

Cited by 22 publications

References 13 publications

Multi-pulse edge-localized states on quantum graphs

Multi-pulse edge-localized states on quantum graphs

Stability theory for the NLS equation on looping edge graphs

Standing waves of the quintic NLS equation on the tadpole graph

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