Strong instability of standing waves for a system NLS with quadratic interaction

Dinh, Van Duong

doi:10.48550/arxiv.1810.00676

Cited by 3 publications

(3 citation statements)

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“…The purpose of this paper is to investigate instability of the standing wave solution to (1.2) of the form u = e iωt φ 1,ω , e 2iωt φ 2,ω , where ω ∈ R and φ ω = (φ 1,ω , φ 2,ω ) is a R 2 -valued function. In the case of (NLS), Hamano [13] proves strong blow-up instability (see Definition 1.3 below) of standing wave solutions in N = 5 by giving a threshold for scattering or blow-up below the ground state (see also Dinh [9]). In [8], Dinh investigates stability of standing solutions for N 3.…”

Section: Introductionmentioning

confidence: 99%

Strong blow-up instability for standing wave solutions to the system of the quadratic nonlinear Klein-Gordon equations

Miyazaki

2019

Preprint

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This paper is concerned with strong blow-up instability (Definition 1.3) for standing wave solutions to the system of the quadratic nonlinear Klein-Gordon equations. In the single case, namely the nonlinear Klein-Gordon equation with power type nonlinearity, stability and instability for standing wave solutions have been extensively studied. On the other hand, in the case of our system, there are no results concerning the stability and instability as far as we know.In this paper, we prove strong blow-up instability for the standing wave to our system. The proof is based on the techniques in Ohta and Todorova [25]. It turns out that we need the mass resonance condition in two or three space dimensions whose cases are the mass-subcritical case.

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Section: Introductionmentioning

confidence: 99%

Strong blow-up instability for standing wave solutions to the system of the quadratic nonlinear Klein-Gordon equations

Miyazaki

2019

Preprint

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“…Also, the scattering below the ground state, in dimension n = 4, was dealt with in [21]. Additional properties of system (1.4) and additional models of two and three wave systems with quadratic nonlinearities can be found in [5], [4], [8], [9], [11], [17], [39], [38], [41] and references therein. Particularly, in [5] is presented an extensive overview about models in χ (2) media; derivation of sets of equations with quadratic nonlinearities from Maxwell's equations is done.…”

Section: Introductionmentioning

confidence: 99%

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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“…Stability of standing waves for dimensions 1 ď n ď 3 was treated in [65] and [18]. The instability of standing waves in dimension n " 5 was considered in [17]. The scattering problem for dimension n " 4 was studied in [37] and for dimension n " 5 in [28].…”

Section: Introductionmentioning

confidence: 99%

On the dynamics of non-linear Schrödinger systems with quadratic-type interactions

Francisco Noguera Salgado

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Nesta tese, analisamos a dinâmica de um sistema acoplado de equações de Schrödinger com iterações de crescimento do tipo quadrático nos espaços L 2 , H 1 e L 2 com peso. Impondo certas condições sobre as não linearidades e os parâmetros do sistema, apresentamos alguns resultados. Primeiro aplicamos o princípio de contração e mostramos a existência de soluções locais. Em seguida, usando as quantidades conservadas, obtemos estimativas a priori que nos permitem estender as soluções globalmente no tempo. Além disso, fazendo uso da abordagem variacional, provamos a existência de soluções ground state para o sistema elíptico associado ao sistema de evolução. Posteriormente, usamos as soluções ground state e estabelecemos um critério sharp para a existência de soluções globais e de soluções com blow-up em tempo finito. Por fim, realizamos o estudo da estabilidade das soluções ground state. Usando o princípio de concentração-compacidade mostramos que para 1 ď n ď 3 as soluções são estáveis. Por sua vez, usando soluções blow-up provamos a instabilidade para n " 4 e n " 5. Finalmente, consideramos o caso não ressonante e determinamos quais dos resultados previos ainda continuam válidos sob esta condição. Além disso, provamos um resultado de blow-up para soluções do sistema em dimensão n " 5, quando o dado inicial é considerado radial.Palavras-chave: Sistema de equações de Schrödinger; Interações de crescimento do tipo quadrático; Boa colocação; Ondas estacionárias; Blow-up; Estabilidade; Instabilidade.

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Strong instability of standing waves for a system NLS with quadratic interaction

Abstract: We study the strong instability of standing waves for a system of nonlinear Schrödinger equations with quadratic interaction under the mass resonance condition in dimension d = 5.

Cited by 3 publications

References 12 publications

Strong blow-up instability for standing wave solutions to the system of the quadratic nonlinear Klein-Gordon equations

Strong blow-up instability for standing wave solutions to the system of the quadratic nonlinear Klein-Gordon equations

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

On the dynamics of non-linear Schrödinger systems with quadratic-type interactions

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