Intermediate long-wave equation on a half-line

Alejandre, Martín Patricio Árciga; Kaikina, Elena I.

doi:10.1007/s00028-011-0109-z

Cited by 7 publications

(1 citation statement)

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“…Initial-boundary value problems. The global well-posedness of the initialboundary value problem the BO and the ILW equation on the half-line with zero boundary condition at x = 0 is proven respectively in [93] and in [25], for small initial data in suitable weighted Sobolev space. Moreover the long time asymptotic is given, for instance, one gets for the ILW equation [25] u(x, t) = 1 3πδt…”

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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Section: Variamentioning

confidence: 99%