Dispersive shock waves in systems with nonlocal dispersion of Benjamin–Ono type

El, Gennady; Nguyen, Lu Trong Khiem; Smyth, Noel F.

doi:10.1088/1361-6544/aaa10a

Cited by 18 publications

(19 citation statements)

References 69 publications

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“…A generic and extensively studied solution of nonlinear, dispersive wave equations, such as the Korteweg-de Vries (KdV), the nonlinear Schrödinger and the Sine-Gordon equations, is the solitary wave, or soliton for integrable equations [1]. Another generic solution of such equations is the dispersive shock wave (DSW), to nonlinear dispersive wave equations with Benjamin-Ono dispersion, for which the equation governing the periodic wave solution is not of the form u 2 θ = r 2 (u)P(u), but the DSW is of KdV type [36]. In many observational measurements only the solitary wave edge of a DSW can be fully resolved [5], so the restriction of El's method to the leading and trailing edges of a DSW is not a severe one.…”

Section: Introductionmentioning

confidence: 99%

Dispersive shock waves governed by the Whitham equation and their stability

Marchant

Smyth

2018

Proc. R. Soc. A.

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Dispersive shock waves (DSWs), also termed undular bores in fluid mechanics, governed by the non-local Whitham equation are studied in order to investigate short wavelength effects that lead to peaked and cusped waves within the DSW. This is done by combining the weak nonlinearity of the Kortewegde Vries equation with full linear dispersion relations. The dispersion relations considered are those for surface gravity waves, the intermediate long wave equation and a model dispersion relation introduced by Whitham to investigate the 120 • peaked Stokes wave of highest amplitude. A dispersive shock fitting method is used to find the leading (solitary wave) and trailing (linear wave) edges of the DSW. This method is found to produce results in excellent agreement with numerical solutions up until the lead solitary wave of the DSW reaches its highest amplitude. Numerical solutions show that the DSWs for the water wave and Whitham peaking kernels become modulationally unstable and evolve into multi-phase wavetrains after a critical amplitude which is just below the DSW of maximum amplitude.

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Section: Introductionmentioning

confidence: 99%

Dispersive shock waves governed by the Whitham equation and their stability

Marchant

Smyth

2018

Proc. R. Soc. A.

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“…The dispersive shock fitting method has been extended and modified to apply to nonlinear dispersive wave equations with Benjamin-Ono dispersion. 30 This dispersive shock fitting method for Benjamin-Ono dispersion again determines the leading and trailing edges of the DSW, which were found to agree with the known DSW solution of the Benjamin-Ono equation, 28 based on its known Whitham modulation equations in Riemann invariant form. 27 As well as arising from step initial conditions, DSWs also arise from the breaking of a smooth initial condition in the small dispersion limit of the governing nonlinear dispersive wave equation, which has been analyzed for the Benjamin-Ono equation, 31,32 based on the similar problem for the KdV equation 33 using the inverse scattering solution of the equation.…”

Section: Introductionmentioning

confidence: 59%

“…The change in the sign of the dispersion, = −1 to = 1, is the key to determining the solitary wave edge of a DSW governed by a nonlinear dispersive wave equation with Benjamin-Ono-type dispersion using the dispersive shock fitting method. 30 Introducing the velocity field = , we can rewrite this bidirectional wave equation as a system of first-order equations as − (10 + ) +  =0, − =0.…”

Section: Dsw Governed By Bbo Equationmentioning

confidence: 99%

“…To solve these ODEs, a boundary condition is needed, which was found from the trailing edge of the BBO DSW as determined using the dispersive shock fitting method, 20,24 extended to deal with Benjamin-Ono dispersion. 30 The determination of the DSW solution of the BBO equation of the present work is then a hybrid of the two standard methods to determine DSW solutions, Whitham modulation theory and dispersive shock fitting. If the Whitham modulation equations for a given nonlinear dispersive wave equation can be determined, it represents a viable method to determine the full DSW structure without the need for a Riemann invariant form of these modulation equations.…”

Section: Introductionmentioning

confidence: 99%

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Dispersive shock waves for the Boussinesq Benjamin–Ono equation

Nguyen

Smyth

2021

Stud Appl Math

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In this work, the dispersive shock wave (DSW) solution of a Boussinesq Benjamin-Ono (BBO) equation, the standard Boussinesq equation with dispersion replaced by nonlocal Benjamin-Ono dispersion, is derived. This DSW solution is derived using two methods, DSW fitting and from a simple wave solution of the Whitham modulation equations for the BBO equation. The first of these yields the two edges of the DSW, while the second yields the complete DSW solution. As the Whitham modulation equations could not be set in Riemann invariant form, the ordinary differential equations governing the simple wave are solved using a hybrid numerical method coupled to the dispersive shock fitting which provides a suitable boundary condition. The full DSW solution is then determined, which is found to be in excellent agreement with numerical solutions of the BBO equation.This hybrid method is a suitable and relatively simple method to fully determine the DSW solution of a nonlinear dispersive wave equation for which the (hyperbolic) Whitham modulation equations are known, but their Riemann invariant form is not.

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“…We refer to [192] for a study of the BO hierarchy with positive initial data in a suitable class in the zero-dispersion limit. We also mention [74] for a description of dispersive shock waves for a class of nonlinear wave equations with a nonlocal BO type dispersion which does not use the integrability of the equation.…”

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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Dispersive shock waves in systems with nonlocal dispersion of Benjamin–Ono type

Cited by 18 publications

References 69 publications

Dispersive shock waves governed by the Whitham equation and their stability

Dispersive shock waves governed by the Whitham equation and their stability

Dispersive shock waves for the Boussinesq Benjamin–Ono equation

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

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