Spectra of Linearized Operators for NLS Solitary Waves

Chang, Shu‐Chiuan; Gustafson, Stephen J.; Nakanishi, Kenji; Tsai, Tai‐Peng

doi:10.1137/050648389

Cited by 145 publications

(163 citation statements)

References 36 publications

(58 reference statements)

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“…Bifurcations of resonances at the endpoints were studied in one dimension using the method of Evans functions (see [24], [18], [19]), numerically in one and two dimensions in ( [9]). The primary goal of the study of such bifurcations is to show the structural instability of such resonances and eigenvalues, that they disappear when generic perturbations are applied to the NLS equation.…”

Section: Assumption 2 the Potentialsmentioning

confidence: 99%

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On Threshold Eigenvalues and Resonances for the Linearized NLS Equation

Vougalter

2010

Math. Model. Nat. Phenom.

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Section: Assumption 2 the Potentialsmentioning

confidence: 99%

“…Spectral properties of the linearized NLS operator were studied recently in (see e.g. [13], [10], [29], [30], [9]). Its spectrum σ(L 0 ) is symmetric with respect to the real and imaginary axes.…”

Section: Introductionmentioning

confidence: 99%

On Threshold Eigenvalues and Resonances for the Linearized NLS Equation

Vougalter

2010

Math. Model. Nat. Phenom.

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“…The resonances at p j can also be explained easily from the complete eigenvalue picture provided in Ref. 9. Namely, for each p j and p close to p j , we have eigenvalues λ(p), so that lim p→ p j λ( p) = 1.…”

Section: Introductionmentioning

confidence: 99%

Conditional stability theorem for the one dimensional Klein-Gordon equation

Demirkaya

Stanislavova

2011

Journal of Mathematical Physics

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The paper addresses the conditional non-linear stability of the steady state solutions of the one-dimensional Klein-Gordon equation for large time. We explicitly construct the center-stable manifold for the steady state solutions using the modulation method of Soffer and Weinstein and Strichartz type estimates. The main difficulty in the onedimensional case is that the required decay of the Klein-Gordon semigroup does not follow from Strichartz estimates alone. We resolve this issue by proving an additional weighted decay estimate and further refinement of the function spaces, which allows us to close the argument in spaces with very little time decay. C

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“…The main difficulty in the application of HUM stems from the lack of self-adjointness of the linearized operator, which makes determining its spectral properties more intricate. The analysis reveals that most spectral properties known to hold for the linearized NLS operator in the whole-space case [6,7], carry over to the zero-boundary case considered in this paper (see Section 3.1.1). § This fact is of independent interest, but it is, to the best of our knowledge, not available in the literature.…”

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confidence: 96%

Controllability of the nonlinear Schrödinger equation in the vicinity of the ground state

Lange

Teismann

2007

Math Methods in App Sciences

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SUMMARYLocal exact controllability of the one-dimensional NLS (subject to zero-boundary conditions) with distributed control is shown to hold in a H 1 -neighbourhood of the nonlinear ground state. The Hilbert Uniqueness Method (HUM), due to Lions, is applied to the linear control problem that arises by linearization around the ground state. The application of HUM crucially depends on the spectral properties of the linearized NLS operator which are given in detail. INTRODUCTIONThe control properties of many partial differential equations (PDEs) arising in physics and engineering have been studied extensively. Those investigations include exact and/or optimal controllability for the (linear and nonlinear) heat, wave, beam and plate equations as well as equations of elasticity and Navier-Stokes equations, to mention just some of the most prominent examples. For Schrödinger equations, however, the control theory is markedly less developed. For a brief survey on control results for (linear and nonlinear) Schrödinger equations, see e.g. [1]. The purpose of this paper is to establish local exact controllability for the one-dimensional nonlinear Schrödinger (NLS) equation (subject to zero-boundary conditions) with distributed control. Specifically, we will consider the following control problem: where g (x) denotes the indicator function for some fixed, possibly small, open subinterval ⊂ := (0, 1) representing the spatial region in which the control is applied. Given a fixed control 'horizon' T >0 and initial and target states y 0 and y 1 , the objective is to construct a control function u = u(t, x) that will steer the state y from y 0 to y 1 , i.e. the unique solution y = y(t, x) of (1a)- (1c) is to satisfy (1d). The control problem (1a)-(1d) was posed in [2,3]. A small-data controllability result for periodic boundary conditions is contained in [4].In this paper, we will concentrate on the special case of the focusing cubic NLS equation, i.e. f (s) =−sMore general nonlinearities could be treated, but this is not our main interest here. So we restrict ourselves to the prototypical cubic nonlinearity. Our main result states that the control problem (1a)-(1d) is soluble locally within an H 1 -neighbourhood of the ground-state solution (t, x) = e i t (x), if the control time T >0 is sufficiently large. Here (x) denotes the (nonlinear time-independent) ground state, ‡ i.e. the (real and) positive solution of the boundary value problemwhich is known to exist and to be unique (see Section A.1). Our main result reads. Theorem 1Let ⊂ (0, 1), >0 be given and let denote the ground state. Then, there exist T >0 and >0 such that, for any y 0 ,The theorem will be proved by applying the implicit function theorem (IFT) to the nonlinear map :where t → y(t; y 0 , u) ∈ C([0, T ]; H 1 0 (0, 1)) denotes the unique solution of (1a)-(1c). To be able to apply the IFT, it will be verified that the linearization * u ( , e i T , 0) : Lions [5]. The main difficulty in the application of HUM stems from the lack of self-adjointness of the linear...

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Spectra of Linearized Operators for NLS Solitary Waves

Cited by 145 publications

References 36 publications

On Threshold Eigenvalues and Resonances for the Linearized NLS Equation

On Threshold Eigenvalues and Resonances for the Linearized NLS Equation

Conditional stability theorem for the one dimensional Klein-Gordon equation

Controllability of the nonlinear Schrödinger equation in the vicinity of the ground state

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