Stability of standing waves for logarithmic Schrodinger equation with attractive delta potential

Pava, Jaime Angulo; Ardila, Alex H.

doi:10.1512/iumj.2018.67.7273

Cited by 16 publications

(22 citation statements)

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“…We also remark that Angulo and Hernandez in [10] showed (via the variational approach) the orbital stability of the ground states ϕ ω,γ in the space W (R) in the case of attractive δ-interaction (γ > 0). In should be noted that investigation of the orbital stability of ϕ ω,γ in the case of repulsive δ-interaction (γ < 0) via constrained minimizer for the action or the energy functional is not applicable (see [10,Remark 4.5]).…”

Section: Previous Results and Basic Notionsmentioning

confidence: 69%

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Stability of standing waves for NLS-log equation with $$\varvec{\delta }$$-interaction

Pava¹,

Goloshchapova²

2017

Nonlinear Differ. Equ. Appl.

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Abstract. We study analytically the orbital stability of the standing waves with a peak-Gausson profile for a nonlinear logarithmic Schrö-dinger equation with δ-interaction (attractive and repulsive). A major difficulty is to compute the number of negative eigenvalues of the linearized operator around the standing wave. This is overcome by the perturbation method, the continuation arguments, and the theory of extensions of symmetric operators.Mathematics Subject Classification (2010). Primary 35Q51, 35J61; Secondary 47E05.

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Section: Previous Results and Basic Notionsmentioning

confidence: 69%

“…In should be noted that investigation of the orbital stability of ϕ ω,γ in the case of repulsive δ-interaction (γ < 0) via constrained minimizer for the action or the energy functional is not applicable (see [10,Remark 4.5]). …”

Section: Previous Results and Basic Notionsmentioning

confidence: 99%

Stability of standing waves for NLS-log equation with $$\varvec{\delta }$$-interaction

Pava¹,

Goloshchapova²

2017

Nonlinear Differ. Equ. Appl.

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“…Equation (1) belongs to a family of models featuring the competition between repulsive (λ 1 ≤ 0)/attractive (λ 1 ≥ 0) and repulsive (λ 2 ≤ 0)/attractive (λ 2 ≥ 0)/terms, that have drawn considerable attention for p = 3 in both the physical and the mathematical communities in recent years (an abridged list of references include [1], [2], [3], [5], [6], [7], [8], [10], [13], [16], [19], [22], [23], [24], [25], [26], [32], [35], [36], [38], [40] and [41]). This combination of nonlinearities in (1) for p > 1 is well-known in optical media (cf.…”

Section: Introductionmentioning

confidence: 99%

Orbital stability of standing waves for the nonlinear Schrödinger equation with attractive delta potential and double power repulsive nonlinearity

Pava,

Melo,

Plaza

2019

Preprint

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In this paper, a nonlinear Schrödinger equation with an attractive (focusing) delta potential and a repulsive (defocusing) double power nonlinearity in one spatial dimension is considered. It is shown, via explicit construction, that both standing wave and equilibrium solutions do exist for certain parameter regimes. In addition, it is proved that both types of wave solutions are orbitally stable under the flow of the equation by minimizing the charge/energy functional.

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“…The case of Schrödinger models on star graphs with δ conditions on the vertex also have been studied recently in Adami et.al ( [1,2] and references therein) and An-gulo&Goloshchapova ( [10,11,12]). The case of either other defect type or nonlinearity have been studied in [3,9,12,13,43]. The specific case λ 1 = 2, λ 2 = −1 and Z > 0 was recently considered by Genoud&Malomed&Weishaupl in [30].…”

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confidence: 99%

On stability properties of the Cubic-Quintic Schródinger equation with <inline-formula><tex-math id="M1">\begin{document}$\delta$\end{document}</tex-math></inline-formula>-point interaction

Pava¹,

Melo²

2019

Communications on Pure &Amp; Applied Analysis

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We study analytically and numerically the existence and orbital stability of the peak-standing-wave solutions for the cubic-quintic nonlinear Schrödinger equation with a point interaction determined by the delta of Dirac. We study the cases of attractive-attractive and attractive-repulsive nonlinearities and we recover some results in the literature. Via a perturbation method and continuation argument we determine the Morse index of some specific self-adjoint operators that arise in the stability study. Orbital instability implications from a spectral instability result are established. In the case of an attractive-attractive case and an focusing interaction we give an approach based in the extension theory of symmetric operators for determining the Morse index. 2000 Mathematics Subject Classification. Primary: 35Q55, 37K40, 37K45; Secondary: 47E05, 35J61. JAIME ANGULO PAVA AND CÉSAR A. HERNÁNDEZ MELO ON STABILITY OF THE CUBIC-QUINTIC SCHRÖDINGER WITH δ 2095 φ ω = φ ω,Z,λ1,λ2 for α = λ1 4 and β = λ2 2096 JAIME ANGULO PAVA AND CÉSAR A. HERNÁNDEZ MELO

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Stability of standing waves for logarithmic Schrodinger equation with attractive delta potential

Cited by 16 publications

References 0 publications

Stability of standing waves for NLS-log equation with $$\varvec{\delta }$$-interaction

Stability of standing waves for NLS-log equation with $$\varvec{\delta }$$-interaction

Orbital stability of standing waves for the nonlinear Schrödinger equation with attractive delta potential and double power repulsive nonlinearity

On stability properties of the Cubic-Quintic Schródinger equation with <inline-formula><tex-math id="M1">\begin{document}$\delta$\end{document}</tex-math></inline-formula>-point interaction

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