Semilinear Schrödinger Equations

Cazenave, Thierry

doi:10.1090/cln/010

Cited by 2,177 publications

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“…In particular, modulo translation and phase, there exists a unique positive solution, which is explicitly known. This solution is even and is a ground state (see, for example, [4,7,29] for such results). For γ = 0, an explicit solution of (3) was presented in [13,22] and the following was proved in [12,13].…”

Section: Notationsmentioning

confidence: 91%

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Instability of bound states of a nonlinear Schrödinger equation with a Dirac potential

Coz

Fukuizumi

Fibich

et al. 2008

Physica D: Nonlinear Phenomena

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We study analytically and numerically the stability of the standing waves for a nonlinear Schrödinger equation with a point defect and a power type nonlinearity. A major difficulty is to compute the number of negative eigenvalues of the linearized operator around the standing waves. This is overcome by a perturbation method and continuation arguments. Among others, in the case of a repulsive defect, we show that the standing-wave solution is stable in H 1 rad (R) and unstable in H 1 (R) under subcritical nonlinearity. Further we investigate the nature of instability: under critical or supercritical nonlinear interaction, we prove the instability by blowup in the repulsive case by showing a virial theorem and using a minimization method involving two constraints. In the subcritical radial case, unstable bound states cannot collapse, but rather narrow down until they reach the stable regime (a finite-width instability). In the nonradial repulsive case, all bound states are unstable, and the instability is manifested by a lateral drift away from the defect, sometimes in combination with a finite-width instability or a blowup instability.

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Section: Notationsmentioning

confidence: 91%

“…The local well-posedness of the Cauchy problem for (2) is ensured by [7,Theorem 4.6.1]. Indeed, the operator −∂ 2 x − γ δ is a self-adjoint operator on L 2 (R) (see [2,Chapter I.3.1] and Section 2 for details).…”

Section: Notationsmentioning

confidence: 99%

Instability of bound states of a nonlinear Schrödinger equation with a Dirac potential

Coz

Fukuizumi

Fibich

et al. 2008

Physica D: Nonlinear Phenomena

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“…The idea of the proof is an adaptation of the proof of [17,Theorem 9.3.4]. The following lemma implies that Q and E are well-defined on W .…”

Section: The Cauchy Problem In Wmentioning

confidence: 99%

“…Indeed, we may check the assumptions of Theorem 3.3.1 in [17]. Note that f n is Lipschitz continuous from R + to R. We also notice that…”

Section: Moreover If An Initial Data U 0 Is Even the Solution U(t) mentioning

confidence: 99%

Stability of standing waves for NLS-log equation with $$\varvec{\delta }$$-interaction

Pava¹,

Goloshchapova²

2017

Nonlinear Differ. Equ. Appl.

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Abstract. We study analytically the orbital stability of the standing waves with a peak-Gausson profile for a nonlinear logarithmic Schrö-dinger equation with δ-interaction (attractive and repulsive). A major difficulty is to compute the number of negative eigenvalues of the linearized operator around the standing wave. This is overcome by the perturbation method, the continuation arguments, and the theory of extensions of symmetric operators.Mathematics Subject Classification (2010). Primary 35Q51, 35J61; Secondary 47E05.

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“…Let us introduce the following linear combination 18) and let [S] K,out be the set of all its values as k * l j ∈ K S , λ ∈ Λ m , namely…”

Section: Resonance Invariant Nk-spectrummentioning

confidence: 99%

Nonlinear Dynamics of a System of Particle-Like Wavepackets

Babin

Figotin

Instability in Models Connected With Fluid Flows I

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This work continues our studies of nonlinear evolution of a system of wavepackets. We study a wave propagation governed by a nonlinear system of hyperbolic PDE's with constant coefficients with the initial data being a multi-wavepacket. By definition a general wavepacket has a well defined principal wave vector, and, as we proved in previous works, the nonlinear dynamics preserves systems of wavepackets and their principal wave vectors. Here we study the nonlinear evolution of a special class of wavepackets, namely particle-like wavepackets. A particle-like wavepacket is of a dual nature: on one hand, it is a wave with a well defined principal wave vector, on the other hand, it is a particle in the sense that it can be assigned a well defined position in the space. We prove that under the nonlinear evolution a generic multi-particle wavepacket remains to be a multi-particle wavepacket with a high accuracy, and every constituting single particle-like wavepacket not only preserves its principal wave number but also it has a well-defined space position evolving with a constant velocity which is its group velocity. Remarkably the described properties hold though the involved single particle-like wavepackets undergo nonlinear interactions and multiple collisions in the space. We also prove that if principal wavevectors of multi-particle wavepacket are generic, the result of nonlinear interactions between different wavepackets is small and the approximate linear superposition principle holds uniformly with respect to the initial spatial positions of wavepackets.

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Semilinear Schrödinger Equations

Cited by 2,177 publications

References 0 publications

Instability of bound states of a nonlinear Schrödinger equation with a Dirac potential

Instability of bound states of a nonlinear Schrödinger equation with a Dirac potential

Stability of standing waves for NLS-log equation with $$\varvec{\delta }$$-interaction

Nonlinear Dynamics of a System of Particle-Like Wavepackets

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