Dispersive shock waves for the Boussinesq Benjamin–Ono equation

Nguyen, Lu Trong Khiem; Smyth, Noel F.

doi:10.1111/sapm.12381

Cited by 11 publications

(4 citation statements)

References 43 publications

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“…He and his collaborators successfully adjusted the fitting method to make it generally applicable to BO systems. The modified DSW fitting method was then applied to the BO equation [103], the Calogero-Sutherland dispersive hydrodynamic system [103], and the Boussinesq-BO equation [104], yielding excellent agreement with full numerical solutions.…”

Section: Dispersive Shock Waves Modulation Theory and Non-convex Disp...mentioning

confidence: 96%

Nematic Dispersive Shock Waves from Nonlocal to Local

et al. 2021

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The structure of optical dispersive shock waves in nematic liquid crystals is investigated as the power of the optical beam is varied, with six regimes identified, which complements previous work pertinent to low power beams only. It is found that the dispersive shock wave structure depends critically on the input beam power. In addition, it is known that nematic dispersive shock waves are resonant and the structure of this resonance is also critically dependent on the beam power. Whitham modulation theory is used to find solutions for the six regimes with the existence intervals for each identified. These dispersive shock wave solutions are compared with full numerical solutions of the nematic equations, and excellent agreement is found.

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Section: Dispersive Shock Waves Modulation Theory and Non-convex Disp...mentioning

confidence: 96%

Nematic Dispersive Shock Waves from Nonlocal to Local

et al. 2021

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“…Thanks to Whitham theory and, when applicable, the inverse scattering transform, much is known about small dispersion limits for (1+1)-dimensional nonlinear wave equations (e.g. see [13,21,25,30,38,45] and references therein). On the other hand, small dispersion limits for (2+1)-dimensional systems have been much less studied and (3+1)-dimensional systems apparently have not been studied at all.…”

Section: Introductionmentioning

confidence: 99%

Whitham modulation theory for the defocusing nonlinear Schrödinger equation in two and three spatial dimensions

Abeya¹,

Biondini²,

Hoefer³

2023

J. Phys. A: Math. Theor.

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The Whitham modulation equations for the defocusing nonlinear Schrödinger (NLS) equation in two and three spatial dimensions are derived using a two-phase ansatz for the periodic traveling wave solutions and by period-averaging the conservation laws of the NLS equation. The resulting Whitham modulation equations are written in vector form, which allows one to show that they preserve the rotational invariance of the NLS equation, as well as the invariance with respect to scaling and Galilean transformations, and to immediately generalize the calculations from two spatial dimensions to three. The transformation to Riemann-type variables is described in detail; the harmonic and soliton limits of the Whitham modulation equations are explicitly written down; and the reduction of the Whitham equations to those for the radial NLS equation is explicitly carried out. Finally, the extension of the theory to higher spatial dimensions is brieﬂy outlined. The multidimensional NLS-Whitham equations obtained here may be used to study large amplitude wavetrains in a variety of applications including nonlinear photonic and matter waves.

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“…Whitham theory does not require integrability of the original PDE, and therefore it can also be applied to non-integrable PDEs. Thanks to Whitham theory and, when applicable, the inverse scattering transform (IST), much is known about small dispersion limits for (1+1)-dimensional nonlinear wave equations (e.g., see [13,21,25,30,38,45] and references therein). On the other hand, small dispersion limits for (2+1)dimensional systems have been much less studied and (3+1)-dimensional systems apparently have not been studied at all.…”

Section: Introductionmentioning

confidence: 99%

Whitham modulation theory for the defocusing nonlinear Schrodinger equation in two and three spatial dimensions

Abeya,

Biondini,

Hoefer

2023

Preprint

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The Whitham modulation equations for the defocusing nonlinear Schrödinger (NLS) equation in two, three and higher spatial dimensions are derived using a two-phase ansatz for the periodic traveling wave solutions and by period-averaging the conservation laws of the NLS equation. The resulting Whitham modulation equations are written in vector form, which allows one to show that they preserve the rotational invariance of the NLS equation, as well as the invariance with respect to scaling and Galilean transformations, and to immediately generalize the calculations from two spatial dimensions to three. The transformation to Riemann-type variables is described in detail; the harmonic and soliton limits of the Whitham modulation equations are explicitly written down; and the reduction of the Whitham equations to those for the radial NLS equation is explicitly carried out. Finally, the extension of the theory to higher spatial dimensions is briefly outlined. The multidimensional NLS-Whitham equations obtained here may be used to study large amplitude wavetrains in a variety of applications including nonlinear photonics and matter waves.

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Dispersive shock waves for the Boussinesq Benjamin–Ono equation

Cited by 11 publications

References 43 publications

Nematic Dispersive Shock Waves from Nonlocal to Local

Nematic Dispersive Shock Waves from Nonlocal to Local

Whitham modulation theory for the defocusing nonlinear Schrödinger equation in two and three spatial dimensions

Whitham modulation theory for the defocusing nonlinear Schrodinger equation in two and three spatial dimensions

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