On the Orbital Stability of Standing-Wave Solutions to a Coupled Non-Linear Klein-Gordon Equation

Garrisi, Daniele

doi:10.1515/ans-2012-0311

Cited by 23 publications

(26 citation statements)

References 9 publications

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“…In [23] the precompactness of minimizing sequences is obtained assuming N = 1. To exclude the dichotomy the authors crucially applied [1, Lemma 2.10] which depends in turn on original ideas introduced in [5], see also [12]. In [1, Lemma 2.10] it is shown that the H 1 (R) norm of some functions are strictly decreasing when the masses of the functions are symmetrically rearranged.…”

Section: Introductionmentioning

confidence: 99%

Existence and orbital stability of standing waves for nonlinear Schrödinger systems

Gou

Jeanjean

2016

Nonlinear Analysis

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In this paper we investigate the existence of solutions in H 1 (R N ) × H 1 (R N ) for nonlinear Schrödinger systems of the formunder the constraints4 N for i = 1, 2 and r 1 +r 2 < 2+ 4 N . This problem is motivated by the search of standing waves for an evolution problem appearing in several physical models. Our solutions are obtained as constrained global minimizers of an associated functional. Note that in the system λ 1 and λ 2 are unknown and will correspond to the Lagrange multipliers. Our main result is the precompactness of the minimizing sequences, up to translation. Assuming the local well posedness of the associated evolution problem we then obtain the orbital stability of the standing waves associated to the set of minimizers.

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Section: Introductionmentioning

confidence: 99%

Existence and orbital stability of standing waves for nonlinear Schrödinger systems

Gou

Jeanjean

2016

Nonlinear Analysis

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“…and the assumption (1.2) made in [8]. In Proposition 3.5 we prove that the coupling term of functions satisfying (1.2) must be non-positive and monotonically non-increasing on each component; we can also show that there are non-linearities satisfying our assumptions (A 0 )-(A 4 ) with a sign-changing coupling term.…”

Section: Introductionmentioning

confidence: 59%

“…We conclude by showing that our assumptions are too general to allow an approach based on the symmetric rearrangement and inequality (1.2), as it has been done in [8] or [2]. We de ne 4 ( , ) := − | | + | | + | | + | | , > 0, where 2 < 2 < 2 < < 2 * and = 2.…”

Section: Proposition 32 There Holds Inf( ) = 2 = Inf(λ)mentioning

confidence: 99%

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Standing-wave solutions to a system of non-linear Klein–Gordon equations with a small energy/charge ratio

Garrisi¹

2014

Advances in Nonlinear Analysis

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“…Moreover, most works treat the problem in which the parameters such as σ 1 , σ 2 , c are being fixed. There are very few papers which deal with the existence problem of prescribed L 2 -norm solutions, for instance, see [1,2,4,17,27] for the results on prescribed L 2 -norm solutions to two-component coupled systems. Up to our knowledge, [8,20] are the only available works which obtain prescribed L 2 -norm solutions for coupled nonlinear systems with three or more equations.…”

Section: Introductionmentioning

confidence: 99%

Well-Posedness for Multicomponent Schrödinger–gKdV Systems and Stability of Solitary Waves with Prescribed Mass

2018

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In this paper we prove the well-posedness issues of the associated initial value problem, the existence of nontrivial solutions with prescribed L 2 -norm, and the stability of associated solitary waves for two classes of coupled nonlinear dispersive equations. The first problem here describes the nonlinear interaction between two Schrödinger type short waves and a generalized Korteweg-de Vries type long wave and the second problem describes the nonlinear interaction of two generalized Kortewegde Vries type long waves with a common Schrödinger type short wave. The results here extend many of the previously obtained results for two-component coupled Schrödinger-Korteweg-de Vries systems. Mathematics Subject Classification. 35Q53, 35Q55, 35B35, 35B65, 35A15.1 2 3 + (R) in the non-resonant case. In [15], Corcho and Linares improved the local well-posedness result obtained in [28] to a larger region of the Sobolev indices. Recently, Wu [30] obtained the best local well-posedness result for the (1 + 1)-component NLS-KdV system in the resonant case. Our aim here is to obtain analogous results to the full system of equations (1.1) and (1.2), considering general power nonlinearities, in the Sobolev spaces of the form H s × H s × H k and H s × H k × H k , respectively.

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On the Orbital Stability of Standing-Wave Solutions to a Coupled Non-Linear Klein-Gordon Equation

Cited by 23 publications

References 9 publications

Existence and orbital stability of standing waves for nonlinear Schrödinger systems

Existence and orbital stability of standing waves for nonlinear Schrödinger systems

Standing-wave solutions to a system of non-linear Klein–Gordon equations with a small energy/charge ratio

Well-Posedness for Multicomponent Schrödinger–gKdV Systems and Stability of Solitary Waves with Prescribed Mass

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