The scattering transform for the Benjamin–Ono equation in the small-dispersion limit

Whitham modulation theory for the two dimensional Benjamin-Ono (2DBO) equation is presented. A system of five quasi-linear first-order partial differential equations is derived. The system describes modulations of the traveling wave solutions of the 2DBO equation. These equations are transformed to a singularity-free hyrdodynamic-like system referred to here as the 2DBO-Whitham system. Exact reductions of this system are discussed, the formulation of initial value problems is considered, and the system is used to study the transverse stability of traveling wave solutions of the 2DBO equation.

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“…When r 1 = r 2 , the PDEs for r 1 and r 2 coincide. As a result, system (30) reduces to the following 4 × 4 system:…”

Section: Iiia Exact Reductions Of the 2dbo-whitham Systemmentioning

confidence: 99%

Whitham modulation theory for the two-dimensional Benjamin-Ono equation

2017

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“…It is plausible that a method capable of treating a general class of rational potentials can subsequently be extended to arbitrary potentials by an appropriate density argument. Moreover, our results allow for the analysis of the scattering data in the zero-dispersion limit ( → 0) [8].…”

Section: Introductionmentioning

confidence: 91%

“…determining a function β : R + → C (independent of x), called the reflection coefficient. We next introduce the "bound state" eigenfunction w + = j (x) ∈ H + (R) satisfying (5) with w 0 = 0 for a given eigenvalue λ = λ j < 0 and normalized by the condition x j (x) → 1 as |x| → ∞ (uniformly for Im{x} ≥ 0), (8) or equivalently, as can be shown asymptotically from (5),…”

Section: Introductionmentioning

confidence: 99%

Direct Scattering for the Benjamin-Ono Equation with Rational Initial Data

Miller

Wetzel

2015

Studies in Applied Mathematics

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“…Using the exact formulas for the scattering data of the BO equation valid for general rational potential with simple poles that they obtained in [190], Miller and Wetzel [189] analyzed rigorously the scattering data in the small dispersion limit, deducing in particular precise asymptotic formulas for the reflection coefficient, the location of the eigenvalues and their density, and the asymptotic dependence of the phase constant associated with each eigenvalue on the eigenvalue itself. Such an analysis seems to be unknown for more general potentials.…”

Section: 2mentioning

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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The scattering transform for the Benjamin–Ono equation in the small-dispersion limit

Cited by 15 publications

References 25 publications

Whitham modulation theory for the two-dimensional Benjamin-Ono equation

Whitham modulation theory for the two-dimensional Benjamin-Ono equation

Direct Scattering for the Benjamin-Ono Equation with Rational Initial Data

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

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