Dispersive analysis of charge transfer models

“…The orbital stability question (for Klein-Gordon, NLS, as well as many other classes of PDE) was addressed by Shatah [46], Shatah, Strauss [47], Weinstein [55], [56], Grillakis, Shatah, Strauss [22], [23] (who developed an "abstract" theory of orbital stability), Grillakis [21], Comech, Pelinovsky [9]. As far as the question of asymptotic stability is concerned (which is much closer to the present paper), see Soffer, Weinstein [49], [50], Buslaev, Perelman [5], [6], Cuccagna [10], Rodnianski, Schlag, Soffer [42], [43], Perelman [35], [36], [37], Fröhlich, Jonsson, Gustafson, Sigal [16], Fröhlich, Tsai, Yau [17], as well as Gang, Sigal [18], [19]. For surveys of some of this material, see Strauss [51], and Sulem, Sulem [52].…”

Section: (T) Is Unique Amongst All Solutions With These Initial Data mentioning

confidence: 69%

Stable manifolds for all monic supercritical focusing nonlinear Schrödinger equations in one dimension

Krieger

Schlag

2006

J. Amer. Math. Soc.

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113

133

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Standing wave solutions of the one-dimensional nonlinear Schrödinger equation \[ i ∂ t ψ + ∂ x 2 ψ = − | ψ | 2 σ ψ i\partial _t \psi + \partial _{x}^2 \psi = -|\psi |^{2\sigma } \psi \] with σ > 2 \sigma >2 are well known to be unstable. In this paper we show that asymptotic stability can be achieved provided the perturbations of these standing waves are small and chosen to belong to a codimension one Lipschitz surface. Thus, we construct codimension one asymptotically stable manifolds for all supercritical NLS in one dimension. The considerably more difficult L 2 L^2 -critical case, for which one wishes to understand the conditional stability of the pseudo-conformal blow-up solutions, is studied in the authors’ companion paper Non-generic blow-up solutions for the critical focusing NLS in 1-d, preprint, 2005.

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“…2 is obtained by substituting the bound for ξ H 1 into second inequality of (37). We obtain the conservative estimate…”

Section: Bound On the Fluctuation And Proof Of Main Theoremmentioning

confidence: 99%

Long‐time dynamics of KdV solitary waves over a variable bottom

Dejak

Sigal

2006

Comm Pure Appl Math

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We study the variable bottom generalized Korteweg-de Vries (bKdV) where f is a nonlinearity and b is a small, bounded and slowly varying function related to the varying depth of a channel of water. Many variable coefficient KdV-type equations, including the variable coefficient, variable bottom KdV equation, can be rescaled into the bKdV. We study the long time behaviour of solutions with initial conditions close to a stable, b = 0 solitary wave. We prove that for long time intervals, such solutions have the form of the solitary wave, whose centre and scale evolve according to a certain dynamical law involving the function b(t, x), plus an H 1 (R)-small fluctuation.

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Dispersive analysis of charge transfer models

Abstract: We prove L p estimates for charge transfer Hamiltonians, including matrix and inhomogeneous generalizations; such equations appear naturally in the study of multi-soliton systems.

Cited by 93 publications

References 35 publications

Spectra of Linearized Operators for NLS Solitary Waves

Spectra of Linearized Operators for NLS Solitary Waves

Stable manifolds for all monic supercritical focusing nonlinear Schrödinger equations in one dimension

Long‐time dynamics of KdV solitary waves over a variable bottom

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