Nonlinear ring waves in a two-layer fluid

Хуснутдинова, К. Р.; Zhang, Xizheng

doi:10.1016/j.physd.2016.02.013

Cited by 17 publications

(34 citation statements)

References 29 publications

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“…Note that the initial condition cannot be imposed at 0 = 0 and then the choice of 0 is arbitrary. There are several such numerical simulations in the literature, often using the KdV solitary wave as the initial condition 9,15,23,28,29 for instance, although other initial conditions have also been used 15,17 for instance. However, there is concern whether such solutions are independent of the choice of 0 .…”

Section: Discussionmentioning

confidence: 99%

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Initial conditions for the cylindrical Korteweg‐de Vries equation

Grimshaw

2019

Stud Appl Math

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In this paper, we find suitable initial conditions for the cylindrical Korteweg-de Vries equation by first solving exactly the initial-value problem for localized solutions of the underlying axisymmetric linear long-wave equation. The far-field limit of the solution of this linear problem then provides, through matching, an initial condition for the cylindrical Korteweg-de Vries equation. This initial condition is associated only with the leading wave front of the farfield limit of the linear solution. The main motivation is to resolve the discrepancy between the exact mass conservation law, and the "mass" conservation law for the cylindrical Korteweg-de Vries equation. The outcome is that in the linear initial-value problem all the mass is carried behind the wave front, and then the "mass" in the initial condition for the cylindrical Korteweg-de Vries equation is zero. Hence, the evolving solution in the cylindrical Korteweg-de Vries equation has zero "mass." This situation arises because, unlike the well-known unidirectional Korteweg-de Vries equation, the solution of the initial-value problem for the axisymmetric linear long-wave problem contains both outgoing and ingoing waves, but in the cylindrical geometry, the latter are reflected at the origin into outgoing waves, and eventually the total outgoing solution is a combination of these and those initially generated. K E Y W O R D S cylindrical korteweg-de vries equation, axisymmetric linear wave equation, korteweg-de vries solitons 176

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Section: Discussionmentioning

confidence: 99%

“…It was first derived in the context of plasma waves, 9 for surface waves, [10][11][12] and then for internal waves. [13][14][15][16][17] Like the KdV equation (2) it is integrable [18][19][20][21][22] and has "soliton" solutions. It also has two conservation laws of physical interest, analogous to (3,4),…”

Section: Introductionmentioning

confidence: 99%

Initial conditions for the cylindrical Korteweg‐de Vries equation

Grimshaw

2019

Stud Appl Math

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“…Higher dimensional (2 + 1) dimensional DSWs governed by NLS-type equations are much more difficult to analyse and need a non-trivial azimuthal vortex structure to be stable. Indeed, solutions of (2 + 1) dimensional DSWs governed by not just NLS-type equations, but any nonlinear wave equation, are an open topic [46,47]. Hence, the optical beam generating the colloid DSW will be assumed to have a plane front.…”

Section: Colloid Equationsmentioning

confidence: 99%

Optical dispersive shock waves in defocusing colloidal media

Marchant

Smyth

2017

Physica D: Nonlinear Phenomena

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The propagation of an optical dispersive shock wave, generated from a jump discontinuity in light intensity, in a defocusing colloidal medium is analysed. The equations governing nonlinear light propagation in a colloidal medium consist of a nonlinear Schrödinger equation for the beam and an algebraic equation for the medium response. In the limit of low light intensity, these equations reduce to a perturbed higher order nonlinear Schrödinger equation. Solutions for the leading and trailing edges of the colloidal dispersive shock wave are found using modulation theory. This is done for both the perturbed nonlinear Schrödinger equation and the full colloid equations for arbitrary light intensity. These results are compared with numerical solutions of the colloid equations. AbstractThe propagation of an optical dispersive shock wave, generated from a jump discontinuity in light intensity, in a defocussing colloidal medium is analysed. The equations governing nonlinear light propagation in a colloidal medium consist of a nonlinear Schrödinger equation for the beam and an algebraic equation for the medium response. In the limit of low light intensity, these equations reduce to a perturbed higher order nonlinear Schrödinger equation. Solutions for the leading and trailing edges of the colloidal dispersive shock wave are found using modulation theory. This is done for both the perturbed nonlinear Schrödinger equation and the full colloid equations for arbitrary light intensity. These results are compared with numerical solutions of the colloid equations.

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“…In [47], Khusnutdinova and Zhang undertake numerical modeling of weakly nonlinear surface and interfacial ring waves in a two-layer fluid within the framework of the recently derived 2+1-dimensional concentric KdV-type equation. The 2D version of the dam-break problem is studied and the obtained numerical solutions are shown to exhibit the formation of concentric DSWs.…”

Section: Fluid Dynamics Applicationsmentioning

confidence: 99%