On the orbital stability of solitary waves for the 2-coupled nonlinear Schrödinger system

Nguyen, Nghiem V.

doi:10.4310/cms.2011.v9.n4.a3

Cited by 22 publications

(30 citation statements)

References 21 publications

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“…As a matter of fact few papers address the issue of compactness of minimizing sequences for systems as (1.1)-(1.2). Moreover in most of them there is either exactly one constraint [6] or the two constraints cannot be chosen independently [21,22,24]. Concerning (1.4) the more complete results seem to be due to [23].…”

Section: Introductionmentioning

confidence: 89%

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: 89%

Existence and orbital stability of standing waves for nonlinear Schrödinger systems

Gou

Jeanjean

2016

Nonlinear Analysis

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“…If (Φ ω 1 , Ψ ω 2 ) is not only a critical point, but in fact a global minimizer of the constrained variational problem for H(u, v), then (1.8) is called a ground-state solution of (1.2). In some cases, namely when p = r = 2q = 4 and under certain conditions on α, β, and τ, it is possible to show further that the ground-state solutions are solitary waves with the usual sech-profile (see, for example, [33,31]). …”

Section: Introductionmentioning

confidence: 99%

“…The Cazenave and Lions method has since been adapted by different authors to prove existence and stability results of a variety of nonlinear dispersive equations (see, for example, [2,3,9,14,31,33]). …”

Section: Introductionmentioning

confidence: 99%

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Stability of solitary-wave solutions of coupled NLS equations with power-type nonlinearities

Bhattarai

2015

Advances in Nonlinear Analysis

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Abstract. This paper proves existence and stability results of solitary-wave solutions of a system of 2-coupled nonlinear Schrödinger equations with power-type nonlinearities arising in several models of modern physics. The existence of vector solitary-wave solutions (i.e, both components are nonzero) is established via variational methods. The set of minimizers is shown to be stable and further information about the structures of this set are given. The results extend stability results previously obtained by Cipolatti and Zumpichiatti [14], Nguyen and Wang [31,32], and Ohta [33].

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“…They study the dynamics of both exact and approximate solutions. For the case α 12 = α 21 the orbital stability of some solitary wave solutions of (1.1) is established in [35] and [37], see also [3,4,38].…”

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confidence: 99%

On the spectral stability of solitary wave solutions of the vector nonlinear Schrödinger equation

Deconinck

Sheils

Nguyen

et al. 2013

J. Phys. A: Math. Theor.

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Abstract. We consider a system of coupled cubic nonlinear Schrödinger (NLS) equationswhere the interaction coefficients α jk are real. The spectral stability of solitary wave solutions (both bright and dark) is examined both analytically and numerically. Our results build on preceding work by Nguyen et al. and others. Specifically, we present closed-form solitary wave solutions with trivial and non-trivial phase profiles. Their spectral stability is examined analytically by determining the locus of their essential spectrum. Their full stability spectrum is computed numerically using a large-period limit of Hill's method. We find that all nontrivial-phase solutions are unstable while some trivial-phase solutions are spectrally stable. To our knowledge, this paper presents the first investigation of the stability of the solitary waves of the coupled cubic NLS equation without the restriction that all components ψ j are proportional to sech.

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On the orbital stability of solitary waves for the 2-coupled nonlinear Schrödinger system

Cited by 22 publications

References 21 publications

Existence and orbital stability of standing waves for nonlinear Schrödinger systems

Existence and orbital stability of standing waves for nonlinear Schrödinger systems

Stability of solitary-wave solutions of coupled NLS equations with power-type nonlinearities

On the spectral stability of solitary wave solutions of the vector nonlinear Schrödinger equation

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