Analytic smoothing effect for a system of nonlinear Schrödinger equations

Hoshino, Gaku; Ozawa, Tohru

doi:10.7153/dea-05-25

Cited by 12 publications

(7 citation statements)

References 24 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In [20] the time decay estimates of small solutions to the systems under the mass resonance condition in 2dimensional space was revised. In [15] was obtained the global existence of analytic solutions in space dimensions n ≥ 3, under the mass resonance condition for sufficiently small Cauchy data with exponential decay.…”

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

View full text Add to dashboard Cite

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.

show abstract

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

View full text Add to dashboard Cite

show abstract

“…The mass-resonance assumption has also been appeared in various works involving two and three-component Schrödinger systems (see for instance [38], [42], [39], [19], [20], [21] and [22] and references therein). When (RC) does not hold a more careful analysis must be performed and we do not know if solutions in Σ with negative energy, for instance, blow-up or not.…”

Section: (H1)mentioning

confidence: 98%

“…(i) In Section 4 we will show that under assumption (2.18), condition (2. 19) is sharp in the sense if inequality has been reversed and the initial data is radial then the solution must blow up in finite time.…”

Section: 4mentioning

confidence: 99%

Blow-up solutions for a system of Schrödinger equations with general quadratic-type nonlinearities in dimensions five and six

Noguera¹,

Pastor²

2020

Preprint

View full text Add to dashboard Cite

In this work, we show the existence of ground state solutions for an l-component system of non-linear Schrödinger equations with quadratic-type growth interactions in the energy-critical case. They are obtained analyzing a critical Sobolev-type inequality and using the concentration-compactness method. As an application, we prove the existence of blow-up solutions of the system without the mass-resonance condition in dimension six (and five), when the initial data is radial.

show abstract

“…For the analyticity of solutions to nonlinear evolution equations see [2,15,17,22,23,24,27]. Especially the analytic smoothing e¤ect in terms of the Galilei generator JðtÞ ¼ x þ it' was studies extensively (see [4,5,6,7,8,9,10,11,12,13,14,18,19,20,21] and reference therein) in the case of the nonlinear Schrö dinger equations with gauge invariant nonlinearity like juj 2l u, l A N. The non gauge invariant nonlinearity u s in (1.1) is not invariant under the Galilei transform, since we see that…”

Section: Introductionmentioning

confidence: 99%

Analyticity of Solutions to the Non Gauge Invariant Schrödinger Equations

Hoshino

Naumkin

2017

Self Cite

View full text Add to dashboard Cite

Abstract. We study the global Cauchy problem for the non gauge invariant Schrö dinger equationsThe application of the Galilei generator for the proof of the analytic smoothing e¤ect of solutions to the Cauchy problem for non gauge invariant Schrö dinger equations involves di‰culties. In this paper we construct analytic solutions to the non gauge invariant Schrö dinger equations in the case of analytic and su‰ciently small initial data. We use the power like analytic spaces and the analytic Hardy spaces as auxiliary analytic spaces characterized by the Galilei generator. Also we show that if the initial data f decay exponentially and are su‰ciently small in an appropriate norm, then the solutions of the Cauchy problem for non gauge invariant Schrö dinger equations exist globally in time and are analytic.

show abstract

Analytic smoothing effect for a system of nonlinear Schrödinger equations

Cited by 12 publications

References 24 publications

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

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

Blow-up solutions for a system of Schrödinger equations with general quadratic-type nonlinearities in dimensions five and six

Analyticity of Solutions to the Non Gauge Invariant Schrödinger Equations

Contact Info

Product

Resources

About

Analytic smoothing effect for a system of nonlinear Schrödinger equations

Cited by 12 publications

References 24 publications

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

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

Blow-up solutions for a system of Schrödinger equations with general quadratic-type nonlinearities in dimensions five and six

Analyticity of Solutions to the Non Gauge Invariant Schr&ouml;dinger Equations

Contact Info

Product

Resources

About

Analyticity of Solutions to the Non Gauge Invariant Schrödinger Equations