Least Energy Solitary Waves for a System of Nonlinear Schrödinger Equations in $${\mathbb{R}^n}$$

Sirakov, Boyan

doi:10.1007/s00220-006-0179-x

Cited by 360 publications

(302 citation statements)

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“…Thus our theorem improves [22, Theorem 2(iv)]. Note that other sufficient assumptions were also given in [22] (see [Proposition 3.7] therein) to guarantee existence of a positive radial ground state of (1). Nevertheless, those assumptions involve heavy notation and were described with quite complex inequalities, which are not easy to verify.…”

supporting

confidence: 64%

“…Recently, (1) has attracted tremendous attention and has been studied extensively from the point of view of physics (see [1,10,11] for instance) as well as mathematics (see [2,3,4,13,14,15,16,18,19,21,22,24,25]). Solutions of (1) correspond to critical points of the energy functional (1) Let…”

mentioning

confidence: 99%

“…Explicit positive ground states can be obtained in this case. According to [22], (u 1 , u 2 ) with u 1 and u 2 defined by…”

mentioning

confidence: 99%

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Ground states of nonlinear Schrödinger systems

Chang

Liu

2009

Proc. Amer. Math. Soc.

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Abstract. This paper concerns the existence of positive radial ground states of the time-independent Schrödinger system ⎧ ⎨where n = 1, 2, 3, λ j > 0 and µ j > 0 for j = 1, 2, and β > 0. A result from Sirakov, Comm. Math. Phys. 271 (2007), 199-221, is improved.Consider the time-independent Schrödinger system (1)where n = 1, 2, 3, λ j > 0 and µ j > 0 for j = 1, 2, and β > 0. Solutions (u 1 (x), u 2 (x)) of (1) correspond to standing wave solutions (e iλ 1 t u 1 (x), e iλ 2 t u 2 (x)) of the timedependent system of 2 coupled nonlinear Schrödinger equationsThe system (2) stems from many physical problems, especially in nonlinear optics and in the Hartree-Fock theory for Bose-Einstein condensates; see, for example, [1,5,8,9,10,11,12,20,23]. Recently, (1) has attracted tremendous attention and has been studied extensively from the point of view of physics (see [1,10,11] for instance) as well as mathematics (see [2,3,4,13,14,15,16,18,19,21,22,24,25]). Solutions of (1) correspond to critical points of the energy functional (1)

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Ground states of nonlinear Schrödinger systems

Chang

Liu

2009

Proc. Amer. Math. Soc.

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“…with β, which contradicts (16). Therefore, there is an a ∈ R N \ {0} such that (y β − x β )/r β → a, and hence the left hand side of (16) also passes to the limit in β, providing (19).…”

Section: Normalization and Blow-upmentioning

confidence: 95%

Uniform Hölder Bounds for Nonlinear Schrödinger Systems with Strong Competition

Noris

Tavares

Terracini

et al. 2009

Comm Pure Appl Math

228

271

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For the positive solutions of the Gross-Pitaevskii systemwe prove that L ∞ -boundedness implies C 0,α -boundedness, uniformly as β → +∞, for every α ∈ (0, 1). Moreover we prove that the limiting profile, as β → +∞, is Lipschitz continuous. The proof relies upon the blow-up technique and the monotonicity formulae by Almgren and Alt-Caffarelli-Friedman. This system arises in the Hartree-Fock approximation theory for binary mixtures of Bose-Einstein condensates in different hyperfine states. Extensions to systems with k > 2 densities are given.MSC : 35B40, 35B45, 35J55.

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“…The system (5) arises in many physical problems, especially in the Hartree-Fock theory and nonlinear optics. We refer to [1,3,6,[8][9][10]13,14] and references therein.…”

Section: Introductionmentioning

confidence: 99%