On Decay Properties of Solutions to the IVP for the Benjamin–Ono Equation

Flores, Cynthia

doi:10.1007/s10884-013-9321-6

Cited by 8 publications

(7 citation statements)

References 21 publications

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“…In [42] it was proved that this is not possible at least for r < 11/2. Next, we consider the local UCP for solutions of the IVP associated to the k-gBO equation (3.4).…”

Section: The Case Of the Benjamin-ono Equationmentioning

confidence: 99%

On unique continuation for non-local dispersive models

Linares¹,

Ponce²

2021

Preprint

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We consider unique continuation properties of solutions to a family of evolution equations. Our interest is mainly on nonlinear non-local models. This class contains the Benjamin-Ono, the Intermediate Long Wave, the Camassa-Holm, the dispersion generalized Benjamin-Ono and non-local Schrödinger equations as well as their generalizations. We shall review, discuss, expand, and comment on several results. In addition, we shall state some open questions concerning these results and their techniques.

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“…In [42] it was proved that this is not possible at least for r < 11/2. Next, we consider the local UCP for solutions of the IVP associated to the k-gBO equation (3.4).…”

Section: The Case Of the Benjamin-ono Equationmentioning

confidence: 99%

On unique continuation for non-local dispersive models

Linares¹,

Ponce²

2021

Preprint

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“…Weighted spaces. The Cauchy problem for the BO equation in weighted spaces is studied in [104,96,75,76,77,78,79] yielding smoothing and unique Figure 3. Solution to the fKdV equation with α = 0.6 for the initial data u 0 = 5sech 2 x continuation properties [105].…”

Section: Variamentioning

confidence: 99%

“…Weighted spaces. The Cauchy problem for the BO equation in weighted spaces is studied in [104,96,75,76,77,78,79] yielding smoothing and unique continuation properties [105]. A result on the propagation of regularity for solutions to the Cauchy problem associated to the Benjamin-Ono equation is proved in [106].…”

Section: Variamentioning

confidence: 99%

Benjamin-Ono and Intermediate Long Wave Equations: Modeling, IST and PDE

Saut

2019

Fields Institute Communications

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This survey article is focused on two asymptotic models for internal waves, the Benjamin-Ono (BO) and Intermediate Long Wave (ILW) equations that are integrable by inverse scattering techniques (IST). After recalling briefly their (rigorous) derivations we will review old and recent results on the Cauchy problem, comparing those obtained by IST and PDE techniques and also results more connected to the physical origin of the equations. We will consider mainly the Cauchy problem on the whole real line with only a few comments on the periodic case. We will also briefly discuss some close relevant problems in particular the higher order extensions and the two-dimensional (KP like) versions of the BO and ILW equations.Remark 3. The above derivation was performed for purely gravity waves. Surface tension effects result in adding a third order dispersive term in the asymptotic models. One gets for instance the so-called Benjamin equation (see [30,31]):where δ > 0 measures the capillary effects. This equation, which is in some sense close to the KdV equation, is not known to be integrable. Its solitary waves, the existence of which was proven in [30,31] by the degree-theoretic approach, present oscillatory tails. We refer to [145] for the Cauchy problem in L 2 and to [12,21] for further results on the existence and stability of solitary wave solutions and to [46] for numerical simulations.In presence of surface tension, the ILW equation has to be modified in the same way. We are not aware of mathematical results on the resulting equation.Remark 4. The BO equation was fully justified in [219] as a model of long internal waves in a two-fluid system by taking into account the influence of the surface

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“…Lately, the idea of improving theorems in weighted Sobolev spaces for the Benjamin-Ono equation, by assuming a higher spacial decay on the initial data, can be found in [9].…”

Section: Introductionmentioning

confidence: 99%

On decay of the solutions for the dispersion generalized-Benjamin-Ono and Benjamin-Ono equations

Cunha¹

2022

Preprint

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We show that uniqueness results of the kind those obtained for KdV and Schrödinger equations ([7], [28]), are not valid for the dispersion generalized-Benjamin-Ono equation in the weighted Sobolev spacesfor appropriated s and r.In particular, we obtain that the uniqueness result proved for the dispersion generalized-Benjamin-Ono equation ([13]), is not true for all pairs of solutions u 1 = 0 and u 2 = 0. To achieve these results we employ the techniques present in our recent work [6]. We also improve some Theorems established for the dispersion generalized-Benjamin-Ono equation and for the Benjamin-Ono equation ([13], [12]).

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On Decay Properties of Solutions to the IVP for the Benjamin–Ono Equation

Cited by 8 publications

References 21 publications

On unique continuation for non-local dispersive models

On unique continuation for non-local dispersive models

Benjamin-Ono and Intermediate Long Wave Equations: Modeling, IST and PDE

On decay of the solutions for the dispersion generalized-Benjamin-Ono and Benjamin-Ono equations

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