Final state problem for a system of nonlinear Schrödinger equations with three wave interaction

Uriya, Kota

doi:10.1007/s00028-015-0297-z

Cited by 11 publications

(7 citation statements)

References 17 publications

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“…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%

“…Mass-resonance appears as a special relation between the masses of the system and it is closely related with the large time behavior of solutions. As pointed out in [42], it was first considered in Klein-Gordon systems. When considering system (1.4), for instance, it is well known that the value of the parameter κ influences the large-time behavior of its solutions, see [25].…”

Section: (H1)mentioning

confidence: 99%

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

Noguera¹,

Pastor²

2020

Preprint

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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.

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Section: (H1)mentioning

confidence: 98%

Section: (H1)mentioning

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

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“…and E is the energy functional (17). In (25), E denotes the Fréchet derivative of E. To simplify notation we set…”

Section: 2mentioning

confidence: 99%

“…The system (2) presents a fundamental model for three-wave multistep cascading solitons in the absence of walk-off (see [15]). From the mathematical point of view, the study of nonlinear quadratic Schrödinger systems with two or three-wave interaction, in one or higher dimension, has attracted the attention of many researchers in recent years (see [1,3,4,5,9,17,22,23,26,33,34], and references therein). Since such systems appear in physical applications, the topics of study are mainly focused on local and global well-posedness of the Cauchy problem, existence and nonlinear/spectral stability of standing or traveling waves, asymptotic behaviour of global solutions and blow up.…”

mentioning

confidence: 99%

On three-wave interaction Schrödinger systems with quadratic nonlinearities: Global well-posedness and standing waves

Pastor¹

2019

Communications on Pure &Amp; Applied Analysis

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Reported here are results concerning the global well-posedness in the energy space and existence and stability of standing-wave solutions for 1dimensional three-component systems of nonlinear Schrödinger equations with quadratic nonlinearities. For two particular systems we are interested in, the global well-posedness is established in view of the a priori bounds for the local solutions. The standing waves are explicitly obtained and their spectral stability is studied in the context of Hamiltonian systems. For more general Hamiltonian systems, the existence of standing waves is accomplished with a variational approach based on the Mountain Pass Theorem. Uniqueness results are also provided in some very particular cases.

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“…For (NLS system) and its related models, there are amount of literature on those themes, such as [31,32,34] for three waves interaction and [12,14,29] for quadratic-type interaction. In [11,16,30], the authors have studied the problems of (NLS system) that arise in 2D.…”

Section: Introductionmentioning

confidence: 99%

Scattering theory For Quadratic Nonlinear Schrödinger System in dimension six

Gao¹,

Meng²,

Xu³

et al. 2021

Preprint

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In this paper, we study the solutions to the energy-critical quadratic nonlinear Schrödinger system in Ḣ1 × Ḣ1 , where the sign of its potential energy can not be determined directly. If the initial data u 0 is radial or nonradial but satisfies the mass-resonance condition, and its energy is below that of the ground state, using the compactness/rigidity method, we give a complete classification of scattering versus blowing-up dichotomies depending on whether the kinetic energy of u 0 is below or above that of the ground state.

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Final state problem for a system of nonlinear Schrödinger equations with three wave interaction

Cited by 11 publications

References 17 publications

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

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

On three-wave interaction Schrödinger systems with quadratic nonlinearities: Global well-posedness and standing waves

Scattering theory For Quadratic Nonlinear Schrödinger System in dimension six

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