Dynamics of some coupled nonlinear Schrödinger systems in

Xu, Guixiang

doi:10.1002/mma.3015

Cited by 11 publications

(18 citation statements)

References 68 publications

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“…Recalling the scattering and blow‐up result of , which established by Xu (Theorem 1.2 of Xu), which is…”

Section: Introductionmentioning

confidence: 99%

“…System is an important model in physics, which is a good approximation describing the propagation of self‐trapped mutually incoherent wave packets in nonlinear optics. For more phsical background of , one can see Xu and references therein. This paper just focuses on the two coupled nonlinear Schrödinger system .…”

Section: Introductionmentioning

confidence: 99%

“…Ambrosetti and Colorado introduced the standing waves for some coupled Schrödinger equations. Xu established the scattering theory of system by concentration compactness method and profile decomposition.…”

Section: Introductionmentioning

confidence: 99%

“…Considering the steady state for the coupled Nonlinear Schrodinger (NLS) system

- normalΔ \overset{Q}{true} + \overset{Q}{true} = false| \overset{Q}{true} |^{2} \overset{Q}{true} .

Colin and Weinstein considered the natural unconstrained minimization among the nontrivial solutions of and obtained the existence and characterization of the ground state

\overset{Q}{true}

(see Colin and Weinstein for details). If we denote

I false(\overset{φ}{true} false) = \frac{false| false| \nabla \overset{φ}{true} ‖_{L^{2}}^{3} false‖ \overset{φ}{true} ‖_{L^{2}}}{false‖ \overset{φ}{true} ‖_{L^{4}}^{4}}, I_{0} = \inf_{\overset{φ}{true} \in H^{1} ∖ false{0 false}} I false(\overset{φ}{true} false),

then, Pohozaev identities for

\overset{Q}{true}

(which arise from multiplying by

\overset{Q}{true}

and

x \cdot \nabla \overset{Q}{true}

and integrating in

R^{3}

) imply that

false‖ \overset{Q}{true} ‖_{L^{2}} false‖ \nabla \overset{Q}{true} ‖_{L^{2}} = \frac{4}{3} I_{0} .

We define some other quantities (refers to previous works for more details). For

\overset{φ}{true} \in H^{1} : = H^{1} false(R^{3} false) \times H^{1} false(R^{3} false)

, we denote the static energy by …”

Section: Introductionmentioning

confidence: 99%

“…The proof of scattering in Xu is mainly based on Kenig‐Merle's compactness‐rigidity argument . In this paper, we apply Dodson‐Murphy's argument, which simplifies the process of the proof for scattering.…”

Section: Introductionmentioning

confidence: 99%

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On dynamics of the system of two coupled nonlinear Schrödinger in

Xia

2019

Math Methods in App Sciences

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In this paper, we give a new proof of the scattering and blow-up theory of the two coupled nonlinear Schrödinger system via establishing the corresponding interaction Morawetz estimate and scattering criterion. The method of this paper simplifies the proof in Xu, and the result of the paper improves the result in Xu. KEYWORDS blowup, coupled NLS system, Morawetz estimate, scattering MSC CLASSIFICATION 35L70; 35Q55

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“…Recalling the scattering and blow‐up result of , which established by Xu (Theorem 1.2 of Xu), which is…”

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

“…Considering the steady state for the coupled Nonlinear Schrodinger (NLS) system

- normalΔ \overset{Q}{true} + \overset{Q}{true} = false| \overset{Q}{true} |^{2} \overset{Q}{true} .

Colin and Weinstein considered the natural unconstrained minimization among the nontrivial solutions of and obtained the existence and characterization of the ground state

\overset{Q}{true}

(see Colin and Weinstein for details). If we denote

I false(\overset{φ}{true} false) = \frac{false| false| \nabla \overset{φ}{true} ‖_{L^{2}}^{3} false‖ \overset{φ}{true} ‖_{L^{2}}}{false‖ \overset{φ}{true} ‖_{L^{4}}^{4}}, I_{0} = \inf_{\overset{φ}{true} \in H^{1} ∖ false{0 false}} I false(\overset{φ}{true} false),

then, Pohozaev identities for

\overset{Q}{true}

(which arise from multiplying by

\overset{Q}{true}

and

x \cdot \nabla \overset{Q}{true}

and integrating in

R^{3}

) imply that

false‖ \overset{Q}{true} ‖_{L^{2}} false‖ \nabla \overset{Q}{true} ‖_{L^{2}} = \frac{4}{3} I_{0} .

We define some other quantities (refers to previous works for more details). For

\overset{φ}{true} \in H^{1} : = H^{1} false(R^{3} false) \times H^{1} false(R^{3} false)

, we denote the static energy by …”

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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On dynamics of the system of two coupled nonlinear Schrödinger in

Xia

2019

Math Methods in App Sciences

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Global dynamics below the ground state for the focusing Schrödinger equation with a potential

Hamano

Ikeda

2019

J. Evol. Equ.

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In this paper we consider the nonlinear Schrödinger system (NLS) with quadratic interaction in five dimensions. We determine the global behavior of the solutions to the system with data below the ground state. Our proof of the scattering result is based on an argument by . In particular, the new part of this paper is to deal with asymmetric interaction. A blowing-up or growing-up result is proved by combining the argument by Du-Wu-Zhang in [6] and a variational characterization of minimizers. Moreover, we show a blowing-up result if the data has finite variance or is radial.

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Scattering for mass-resonance nonlinear Schrödinger system in 5D

Meng

2021

Journal of Differential Equations

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Dynamics of some coupled nonlinear Schrödinger systems in

Cited by 11 publications

References 68 publications

On dynamics of the system of two coupled nonlinear Schrödinger in

On dynamics of the system of two coupled nonlinear Schrödinger in

Global dynamics below the ground state for the focusing Schrödinger equation with a potential

Scattering for mass-resonance nonlinear Schrödinger system in 5D

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