On a nonlinear Schrödinger system arising in quadratic media

Corcho, Adán J.; Correia, Simão; Oliveira, Filipe; Silva, Jorge Drumond

doi:10.4310/cms.2019.v17.n4.a5

Cited by 9 publications

(8 citation statements)

References 10 publications

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“…4 Q(φ, ψ)K(φ, ψ), which in turn shows that (1.11) holds. This is in agreement with the results in [9] where the blow up was shown if the initial energy is negative (see also [4]). However, since (5/4) 4 > 1, our result is stronger than the one [9].…”

Section: 1supporting

confidence: 93%

“…From the mathematical point of view, the study of nonlinear Schrödinger systems with quadratic interaction has been increasing in recent years. To cite a few, we refer the reader to [3], [4], [7], [8], [9], [12], [14], [15], [20], and references therein. An almost complete study of system (1.1) in L 2 (R n ) and H 1 (R n ) was undertaken in [9] (see also [7], [8]).…”

Section: Introductionmentioning

confidence: 99%

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On the dynamics of a quadratic Schrödinger system in dimension $n = 5$

Noguera

Pastor

2020

Dynamics of Partial Differential Equations

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In this work we give a sharp criterion for the global well-posedness, in the energy space, for a system of nonlinear Schrödinger equations with quadratic interaction in dimension n = 5. The criterion is given in terms of the charge and energy of the ground states associated with the system, which are obtained by minimizing a Weinstein-type functional. The main result is then obtained in view of a sharp Gagliardo-Nirenberg-type inequality.

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Section: 1supporting

confidence: 93%

Section: Introductionmentioning

confidence: 99%

On the dynamics of a quadratic Schrödinger system in dimension $n = 5$

Noguera

Pastor

2020

Dynamics of Partial Differential Equations

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“…In [27] was obtained conditions for the existence of multipulses as well as a description of their geometry. Also, on a recent paper [11] the authors obtained formation of singularities and blow-up in the L 2 (R n )-(super)critical case and derived several stability results concerning the ground state solutions of this system. In [14] Hayashi, Li and Ozawa studied the scattering theory for the system.…”

Section: Results On R N and Tmentioning

confidence: 99%

The nonlinear Quadratic Interactions of the Schrödinger type on the half-line

Barbosa¹,

Cavalcante²

2021

Preprint

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In this work we study the initial boundary value problem associated with the coupled Schrödinger equations with quadratic nonlinearities, that appears in nonlinear optics, on the half-line. We obtain local well-posedness for data in Sobolev spaces with low regularity, by using a forcing problem on the full line with a presence of a forcing term in order to apply the Fourier restriction method of Bourgain. The crucial point in this work is the new bilinear estimates on the classical Bourgain spaces X s,b with b < 1 2 , jointly with bilinear estimates in adapted Bourgain spaces that will used to treat the traces of nonlinear part of the solution. Here the understanding of the dispersion relation is the key point in these estimates, where the set of regularity depends strongly of the constant a measures the scaling-diffraction magnitude indices.

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“…The proof can be performed adapting the arguments of Proposition 6.5.1 in [7]. Nevertheless, we adopt the technique presented in [10], which explores the Hamiltonian structure of the system. The Hamiltonian form of system (1.1) is given by…”

Section: Global Solutions Versus Blow-upmentioning

confidence: 99%

“…This means there exists a constant µ > 0 such thatQ(ψ) = µ, for any ψ ∈ G(1, 0).We will show that for this constant, the sets A µ and G(1, 0) are the same. The proof follows the ideas presented in[10, Lemma 4.2 ]. Lemma 6.13.…”

mentioning

confidence: 94%

On a system of Schrödinger equations with general quadratic-type nonlinearities

Noguera¹,

Pastor²

2019

Preprint

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In this work we study a system of Schrödinger equations involving nonlinearities with quadratic growth. We establish sharp criterion concerned with the dichotomy global existence versus blow-up in finite time. Such a criterion is given in terms on the ground state solutions associated with the corresponding elliptic system, which in turn are obtained by applying variational methods. By using the concentration-compactness method we also investigate the nonlinear stability/instability of the ground states. Contents 1. Introduction 1 2. Preliminaries 4 3. Local and global well-posedness in L 2 and H 1 9 3.1. Local existence of L 2 -solutions 10 3.2. Local existence of H 1 -solutions 12 3.3. Global solutions 13 4. Existence of ground state solutions 17 5. Global solutions versus blow-up 23 5.1. Virial Identities 23 5.2. Global existence in H 1 27 5.3. Blow-up results 28 6. Stability and instability of standing waves 32 6.1. Stability 32 6.2. Instability 45 Acknowledgement 49 References 49

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On a nonlinear Schrödinger system arising in quadratic media

Cited by 9 publications

References 10 publications

On the dynamics of a quadratic Schrödinger system in dimension $n = 5$

On the dynamics of a quadratic Schrödinger system in dimension $n = 5$

The nonlinear Quadratic Interactions of the Schrödinger type on the half-line

On a system of Schrödinger equations with general quadratic-type nonlinearities

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