Formation of the undular bores in shallow water generalized Kaup–Boussinesq model

Gong, Ruizhi; Wang, Deng‐Shan

doi:10.1016/j.physd.2022.133398

Cited by 15 publications

(7 citation statements)

References 31 publications

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“…(J [9] " 3dÑ š‚5Å½™• §£ã 1AE .¥ §1f6Nüz ˜ ‡wÍA ´/¤ÚÑÀÂ Å [10][11][12][13][14][15] §ù˜nØ Ù¥˜ ‡ † A^´¹l¯K" 3íNÄåAEÚ²;ÀÂÅ(Classical Shock Wave, CSW) nØ µeS §¹l¯Käk²;)" ¹lØ Ù˜ý•Ž íNž §²;ÀÂÅÑy § £ ¹lò )²w DÕÅ£Rarefaction Wave, RW¤ " AO/ §3ïÄ1f6N [¯Kž §Xu Gang < [16] 3¢ þ•* NõÛ© y-" ¯¢þ §Cc5ÃØ´nØþ â» [17][18][19][20] "´¢ ¥ #u y §ÑOEOEíÄ š‚5‰AE+• ¯"uÐ [21] " d §'uWhithamN›nØ Ù¦•#ïÄ?Ð § OE"ë•©z [22]- [27]"…”

Section: ¬3k•˜m¥ñymentioning

confidence: 99%

Whitham modulation theory of defocusing nonlinear Schrödinger equation and the classification and evolutions of solutions with initial discontinuity

Rui-Zhi¹,

Wang²

2023

Acta Phys. Sin.

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Since the Whitham modulation theory was first proposed in 1965, it has been widely concerned because of its superiority in studying dispersive fluid dynamics and dealing with discontinuous initial value problems. In this paper, the Whitham modulation theory of the defocusing nonlinear Schrödinger equation is developed, the classification and evolution of the solutions of discontinuous initial value problem are studied. Moreover, the region of dispersive shock wave, the region rarefaction wave, the region of unmodulated wave and the plateau region are distinguished. Particularly, the correctness of the results is verified by direct numerical simulation. Specifically, the solutions of 0-phase and 1-phase and their corresponding Whitham equations are derived by the finite gap integration method. Also the Whitham equation of genus <i>N</i> corresponding to the <i>N</i>-phase periodic wave solution is derived. The basic structures of rarefaction wave and dispersive shock wave are given, in which the boundaries of the regions are calculated in detail. The Riemann invariants and density distributions of dispersive fluids in each case are discussed. When the initial value is fixed as a special one, the vacuum point is considered and analyzed in detail. In addition, the oscillating front and the soliton front in the dispersive shock wave are considered. In fact, the Whitham modulation theory has many wonderful applications in real physics and engineering. The dam problem is investigated as a special Riemann problem, the piston problem of dispersive fluid is analyzed, and the novel undular bores are found.

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Section: ¬3k•˜m¥ñymentioning

confidence: 99%

Whitham modulation theory of defocusing nonlinear Schrödinger equation and the classification and evolutions of solutions with initial discontinuity

Rui-Zhi¹,

Wang²

2023

Acta Phys. Sin.

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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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“…For more than 40 years, the study of predecessors on the Kaup-Boussinesq system has covered a wide range of felds and made many achievements. Especially, the study of traveling wave solutions of the Kaup-Boussinesq system have become a very important research feld [1][2][3][4][5][6]. Many important methods are also used to construct traveling wave solutions [7][8][9][10][11][12] and optical soliton solutions [13][14][15][16][17] of the Kaup-Boussinesq system.…”

Section: Introductionmentioning

confidence: 99%

Traveling Wave Solution of the Kaup–Boussinesq System with Beta Derivative Arising from Water Waves

Chen

2022

Discrete Dynamics in Nature and Society

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The main purpose of this paper is to construct the traveling wave solution of the Kaup–Boussinesq system with beta derivative arising from water waves. By using the complete discriminant system method of polynomial, the rational function solution, the trigonometric function solution, the exponential function solution, and the Jacobian function solution of the Kaup–Boussinesq system with beta derivative are obtained. In order to further explain the propagation of the Kaup–Boussinesq system with beta derivative in water waves, we draw its three-dimensional diagram, two-dimensional diagram, density plot, and contour plot by using Maple software.

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Formation of the undular bores in shallow water generalized Kaup–Boussinesq model

Cited by 15 publications

References 31 publications

Whitham modulation theory of defocusing nonlinear Schrödinger equation and the classification and evolutions of solutions with initial discontinuity

Whitham modulation theory of defocusing nonlinear Schrödinger equation and the classification and evolutions of solutions with initial discontinuity

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

Traveling Wave Solution of the Kaup–Boussinesq System with Beta Derivative Arising from Water Waves

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