Strong Instability of Standing Waves for the Nonlinear Schrödinger Equation in Trapped Dipolar Quantum Gases

Feng, Binhua; Wang, Qingxuan

doi:10.1007/s10884-020-09881-0

Cited by 15 publications

(4 citation statements)

References 36 publications

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“…Next, we consider the strong instability of standing waves for (3) with b = 0 and λ 1 = 0. In this case, the function f (λ) = S b,ω (v λ ) with v λ (x) := λ of NLS has been investigated in similar case, see [20,27,24,25,26,37,36,38,42]. Moreover, the action functionals S ω (v) in these papers only include four terms.…”

Section: For Any ρ > λmentioning

confidence: 99%

“…Note that since the action functional for (4) includes six terms with different scaling rates, there are several essential difficulties in the analyses. Recall that action functionals only including four terms, which corresponds on some of the coefficients in (4) vanishing, the standing waves involved normalized solutions and stability/instability, have been studied by [3,4,5,8,9,16,19,20,24,25,26,36,37,38,42]. In these studies, we see particularly that there are some essential differences between b = 0 and b = 0, where the latter is more difficult to be treated in analysis.…”

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confidence: 99%

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On the standing waves for the X-ray free electron laser Schrödinger equation

Cao¹,

Feng²,

Luo³

2022

DCDS

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In this paper, we are concerned with the standing waves for the following nonlinear Schrödinger equation<disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ i\partial_{t}\psi = -\Delta \psi+b^2(x_1^2+x_2^2)\psi+\frac{\lambda_1}{|x|}\psi+ \lambda_2(|\cdot|^{-1}\ast |\psi|^2)\psi- \lambda_3|\psi|^p \psi,\; \; \; (t,x)\in \mathbb{R}^+\times \mathbb{R}^3, $\end{document} </tex-math></disp-formula>where <inline-formula><tex-math id="M1">\begin{document}$ 0<p<4 $\end{document}</tex-math></inline-formula>. This equation arises as an effective single particle model in X-ray Free Electron Lasers. We mainly study the existence and stability/instability properties of standing waves for this equation, in two cases: the first one is that no magnetic potential is involved (i.e. <inline-formula><tex-math id="M2">\begin{document}$ b = 0 $\end{document}</tex-math></inline-formula> in the equation), and the second one is that <inline-formula><tex-math id="M3">\begin{document}$ b\neq 0 $\end{document}</tex-math></inline-formula>. To be precise, in the first case, when <inline-formula><tex-math id="M4">\begin{document}$ p\in [\frac{4}{3},4) $\end{document}</tex-math></inline-formula>, by considering a minimization problem on a suitable Pohozaev manifold, we prove the existence of radial ground states, and show further that the corresponding standing waves are strongly unstable by blow-up in a finite time. Moreover, by making use of the ideas of these proofs, we are able to prove the existence of normalized solutions, whose proof seems to be new, compared with the studies of normalized solutions in the existing literature. This study also indicates that there is a close connection between the study of the strong instability and the one of the existence of normalized solutions. In the second case, the situation is more difficult to be treated, due to the additional term of the partial harmonic potential. We manage to prove the existence of stable standing waves for <inline-formula><tex-math id="M5">\begin{document}$ p\in (0,4) $\end{document}</tex-math></inline-formula>, where solutions are obtained as global minimizers when <inline-formula><tex-math id="M6">\begin{document}$ p\in (0,\frac{4}{3}] $\end{document}</tex-math></inline-formula>, and as local minimizers when <inline-formula><tex-math id="M7">\begin{document}$ p\in [\frac{4}{3}, 4) $\end{document}</tex-math></inline-formula>. In the mass-critical and supercritical cases <inline-formula><tex-math id="M8">\begin{document}$ p\in [\frac{4}{3}, 4) $\end{document}</tex-math></inline-formula>, we also establish the variational characterization of ground state solutions on a new manifold which is neither of the Nehari type nor of the Pohozaev type, and then prove the existence of ground states. Finally under some assumptions on the coefficients, we prove that the ground state standing waves are strongly unstable.

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Section: For Any ρ > λmentioning

confidence: 99%

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confidence: 99%

On the standing waves for the X-ray free electron laser Schrödinger equation

Cao¹,

Feng²,

Luo³

2022

DCDS

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“…The applications of the Schrödinger equation also include plasma physics [16,17], chemistry [18,19], fluid dynamics [20], quantum [21] and water waves [22,23]. The number of studies in optics has dramatically increased in recent years [24,25].…”

Section: Introductionmentioning

confidence: 99%

Optical solitons and exact solutions of the (2+1) dimensional conformal time fractional Kundu-Mukherjee-Naskar equation via novel extended techniques

Gupta

Yadav

2023

Phys. Scr.

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The current work uses a (2+1) dimensional conformal time fractional Kundu-Mukherjee-Naskar (KMN) modelto investigate optical soliton transmission across an optical fiber that maintains polarization. Three constructivetechniques, namely, the extended power series solution, the new generalized method, and the extendedsinh-Gordon expansion method are utilized to find the exact soliton solutions of this model. The invariantanalysis has been performed on the (2+1) dimensional time fractional KMN model by using the conformaltime fractional derivative. The symmetries obtained using conformal fractional derivative are compared withthe symmetries obtained for integer order KMN model because symmetries using Riemann Liouville fractionalderivative turned out to be trivial. The given system of fractional PDEs has been reduced by using differentialinvariants obtained from various linear combinations of vector fields associated with the infinitesimal generatorof symmetry transformations. These reduced systems of equations are then investigated for their exactsolutions

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“…Because many nonlinear Schrödinger equations enjoy the conservations of mass and energy (see (19) and (20) for (1)), it is particularly relevant to consider normalized solutions in physics. From the mathematical perspective, the variational characterization of normalized solutions can give a better insight for the orbital stability/instability of standing waves, see [6,7,14,23,25,35,51]. Indeed, by considering normalized solutions of NLS, Cazenave and Lions established a general framework for studying the orbital stability.…”

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confidence: 99%

Existence, stability and asymptotic behaviour of normalized solutions for the Davey-Stewartson system

Feng

Liu

2022

DCDS

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In this paper, we undertake a comprehensive study for existence, stability and asymptotic behaviour of normalized solutions for the Davey-Stewartson system<disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ -\Delta u+\omega u = a|u|^{p}u +E_1(|u|^{2})u\; \; \; in\; \mathbb{R}^2\; or \; \mathbb{R}^3,\;\;\;\;\;\;{\rm{(DS)}} $\end{document} </tex-math></disp-formula>which appears in the description of the evolution of surface water waves. In the case of <inline-formula><tex-math id="M7">\begin{document}$ L^2 $\end{document}</tex-math></inline-formula>-critical case, i.e., <inline-formula><tex-math id="M8">\begin{document}$ N = 2 $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M9">\begin{document}$ a>0 $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M10">\begin{document}$ 0<p<2 $\end{document}</tex-math></inline-formula>, we show that normalized ground states blow up as <inline-formula><tex-math id="M11">\begin{document}$ c \nearrow c^*: = \|R\|^2_{L^2} $\end{document}</tex-math></inline-formula>, where <inline-formula><tex-math id="M12">\begin{document}$ R $\end{document}</tex-math></inline-formula> is the ground state solution to equation (DS) with <inline-formula><tex-math id="M13">\begin{document}$ a = 0 $\end{document}</tex-math></inline-formula>. We then give a detailed description for the asymptotic behavior of normalized ground states as <inline-formula><tex-math id="M14">\begin{document}$ c \nearrow c^* $\end{document}</tex-math></inline-formula>. In the case of <inline-formula><tex-math id="M15">\begin{document}$ L^2 $\end{document}</tex-math></inline-formula>-supercritical case, i.e., <inline-formula><tex-math id="M16">\begin{document}$ N = 3 $\end{document}</tex-math></inline-formula>, we prove several existence and stability/instability results. We also give new criteria about global existence and blow-up for the associated evolutional equation.

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Strong Instability of Standing Waves for the Nonlinear Schrödinger Equation in Trapped Dipolar Quantum Gases

Cited by 15 publications

References 36 publications

On the standing waves for the X-ray free electron laser Schrödinger equation

On the standing waves for the X-ray free electron laser Schrödinger equation

Optical solitons and exact solutions of the (2+1) dimensional conformal time fractional Kundu-Mukherjee-Naskar equation via novel extended techniques

Existence, stability and asymptotic behaviour of normalized solutions for the Davey-Stewartson system

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