On the change of amplitude of interacting solitary waves

Byatt-Smith, J. G. B.

doi:10.1017/s0022112087002428

Cited by 20 publications

(12 citation statements)

References 15 publications

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“…To find the change in solitary wave amplitude these equations must be integrated from an initial time, well before the solitary waves interact, to a final time, well after interaction. See Byatt-Smith [7] or Myint and Grimshaw [18] for previous applications.…”

Section: The Solitary Wave Interaction At Third Ordermentioning

confidence: 99%

“…where in this section the scaling used corresponds to that of the KdV system (6)- (7). Substituting (21) into (20) gives the left wave as…”

Section: The Left-moving Wavementioning

confidence: 99%

“…The second-order KdV equation ((1) to O(α)) has been derived with the coefficients (2) by Marchant and Smyth [6] to enable resonant flow over topography to be modeled more accurately and by Byatt-Smith [7] to examine second-order solitary wave interactions. The nature of the solitary wave interaction for higher order KdV models has been considered by a number of authors, both theoretically and numerically.…”

Section: Introductionmentioning

confidence: 99%

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High‐Order Interaction of Solitary Waves on Shallow Water

Marchant

2002

Stud Appl Math

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The interaction of solitary waves on shallow water is examined to fourth order. At first order the interaction is governed by the Korteweg-de Vries (KdV) equation, and it is shown that the unidirectional assumption, of right-moving waves only, is incompatible with mass conservation at third order. To resolve this, a mass conserving system of KdV equations, involving both right-and left-moving waves, is derived to third order. A fourth-order interaction term, in which the right-and left-moving waves are coupled, is also derived as this term is crucial in determining the fourth-order change in solitary wave amplitude. The form of the unidirectional KdV equation is also discussed with nonlocal terms derived at fourth order. The solitary wave interaction is examined using the inverse scattering method for perturbed KdV equations. Central to the analysis at fourth order is the left-moving wave, for which the solution, in integral form, is derived. A symmetry property for the left-moving wave is found, which is used to show that no change in solitary wave amplitude occurs to fourth order. Hence it is concluded that, for surface waves on shallow water, the change in solitary wave amplitude is of fifth order.

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Section: The Solitary Wave Interaction At Third Ordermentioning

confidence: 99%

“…where in this section the scaling used corresponds to that of the KdV system (6)- (7). Substituting (21) into (20) gives the left wave as…”

Section: The Left-moving Wavementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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High‐Order Interaction of Solitary Waves on Shallow Water

Marchant

2002

Stud Appl Math

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“…Extended fifth-order Korteweg-de Vries equations have been considered in [6,9,10,16,17,23]. This equation has been used to describe chains of coupled nonlinear oscillators [25] and most notably gravity-capillary shallow water waves [4,14,30].…”

Section: Introduction and Statement Of The Main Resultsmentioning

confidence: 99%

Limit cycles for a class of fourth-order differential equations

Llibre

Sellami

Makhlouf

2009

Applicable Analysis

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“…Moreover, some contradictions are found in the literature. Some authors claim that the solitary waves suffer from permanent loss of amplitude after the interaction [5,6,7]. The solitary waves do not recover from a transitional loss of amplitude and thus do not return to their nearly original heights.…”

Section: Introductionmentioning

confidence: 99%