Global Well-Posedness, Blow-Up and Stability of Standing Waves for Supercritical NLS with Rotation

Ardila, Alex H.; Hajaiej, Hichem

doi:10.1007/s10884-021-09976-2

Cited by 8 publications

(11 citation statements)

References 20 publications

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“…with L Ω (φ) := −Ω R d φ(x)L z φ(x)dx, is called as the ground state, which will be denoted as φ g . Such definition of the ground state here follows [1,9,29,30,49]. Apart from the ground state, the other nontrivial solution φ of (1.1) is therefore a kind of 'excited state', i.e., S Ω,ω (φ) > S Ω,ω (φ g ), which is referred as the bound state in the literature [9,10].…”

Section: Introductionmentioning

confidence: 99%

“…Apart from the ground state, the other nontrivial solution φ of (1.1) is therefore a kind of 'excited state', i.e., S Ω,ω (φ) > S Ω,ω (φ g ), which is referred as the bound state in the literature [9,10]. Under the focusing nonlinearity (β < 0), the existence of the ground state has been established in [9,29] for the non-rotating (Ω = 0) case of (1.1), and recently in [1] for the rotating (Ω = 0) case. For the non-rotating case, the ground state is found as a positive, smooth and exponentially localized function in space.…”

Section: Introductionmentioning

confidence: 99%

“…As far as we know, on the one hand, stable standing waves are useful in applications and the stability is mathematically relevant to many physical phenomena [5,29,58]. Therefore, under different parameter regimes, i.e., the range of parameters p, d, Ω, ω, many efforts have been devoted to analyzing the stability and instability of the ground state [1,21,29,30,49,57,58] and also the vortices bound state [45]. The existing theoretical results are yet to cover all the parameter regimes, and so direct numerical simulations would be helpful.…”

Section: Introductionmentioning

confidence: 99%

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Computing the action ground state for the rotating nonlinear Schrödinger equation

Liu¹,

Yuan²,

Zhao³

2022

Preprint

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We consider the computations of the action ground state for a rotating nonlinear Schrödinger equation. It reads as a minimization of the action functional under the Nehari constraint. In the focusing case, we identify an equivalent formulation of the problem which simplifies the constraint. Based on it, we propose a normalized gradient flow method with asymptotic Lagrange multiplier and establish the energy-decaying property. Popular optimization methods are also applied to gain more efficiency. In the defocusing case, we prove that the ground state can be obtained by the unconstrained minimization. Then the direct gradient flow method and unconstrained optimization methods are applied. Numerical experiments show the convergence and accuracy of the proposed methods in both cases, and comparisons on the efficiency are discussed. Finally, the relation between the action and the energy ground states are numerically investigated.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Computing the action ground state for the rotating nonlinear Schrödinger equation

Liu¹,

Yuan²,

Zhao³

2022

Preprint

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“…In the mass-supercritical case, i.e., 1 + 4 N < p < 1 + 4 N −2 , a standard scaling argument shows that the energy functional is no longer bounded from below on S(c). Inspired by an idea of Bellazzini-Boussaïd-Jeanjean-Visciglia [9], recent works [5,33] study the local minimization problem: for c, m > 0,…”

mentioning

confidence: 99%

“…Remark 1.1. The results in [4,5,8,32,33] were stated for 0 < Ω < min 1≤j≤N γ j . However, after a careful look (see Lemma 2.1), we can prove the following equivalent norm…”

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confidence: 99%

Existence and stability of standing waves for nonlinear Schrödinger equations with a critical rotational speed

Dinh

2022

Preprint

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We study the existence and stability of standing waves associated to the Cauchy problem for the nonlinear Schrödinger equation (NLS) with a critical rotational speed and an axially symmetric harmonic potential. This equation arises as an effective model describing the attractive Bose-Einstein condensation in a magnetic trap rotating with an angular velocity. By viewing the equation as NLS with a constant magnetic field and with (or without) a partial harmonic confinement, we establish the existence and orbital stability of prescribed mass standing waves for the equation with masssubcritical, mass-critical, and mass-supercritical nonlinearities. Our result extends a recent work of [Bellazzini-Boussaïd-Jeanjean-Visciglia, Comm. Math. Phys. 353 (2017), no. 1, 229-251], where the existence and stability of standing waves for the supercritical NLS with a partial confinement were established.is the harmonic potential with trap frequencies γ j > 0 for j = 1, 2, 3. The angular momentum operatorwith a rotational speed Ω > 0.

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Existence and stability of standing waves for nonlinear Schrödinger equations with a critical rotational speed

Dinh

2022

Lett Math Phys

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Global Well-Posedness, Blow-Up and Stability of Standing Waves for Supercritical NLS with Rotation

Cited by 8 publications

References 20 publications

Computing the action ground state for the rotating nonlinear Schrödinger equation

Computing the action ground state for the rotating nonlinear Schrödinger equation

Existence and stability of standing waves for nonlinear Schrödinger equations with a critical rotational speed

Existence and stability of standing waves for nonlinear Schrödinger equations with a critical rotational speed

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