Solitary waves in media with dispersion and dissipation (a review)

Ильичев, А. Т.

doi:10.1007/bf02831423

Cited by 29 publications

(13 citation statements)

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“…Readers interested in those topics are referred to [1 -7]. On the other hand, for the gravity-capillary case, the experimental data and theoretical results are relatively hard to be compared [8,9].Among the intesting ones, the Hȃrȃgus-CourcelleIl'ichev model for the gravity-capillary and flexuralgravity waves,has recently been proposed [4,5,10], which is (2+1)-dimensional and 6th-ordered, where x, y and t are dimensionless spatial and temporal variables and u is a dimensionless surface deviation. Equation (1) generalizes the Kadomtsev-Petviashvili (KP) equation to the presence of higher order dispersive effects (caused either by sea ice or to surface tension), and the fifth order 0932-0784 / 04 / 1200-0997 $ 06.00 c 2004 Verlag der Zeitschrift für Naturforschung, Tübingen · http://znaturforsch.com Korteweg-de Vries (KdV) equation to (2+1) dimensions.…”

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confidence: 99%

“…has recently been proposed [4,5,10], which is (2+1)-dimensional and 6th-ordered, where x, y and t are dimensionless spatial and temporal variables and u is a dimensionless surface deviation. Equation (1) generalizes the Kadomtsev-Petviashvili (KP) equation to the presence of higher order dispersive effects (caused either by sea ice or to surface tension), and the fifth order 0932-0784 / 04 / 1200-0997 $ 06.00 c 2004 Verlag der Zeitschrift für Naturforschung, Tübingen · http://znaturforsch.com Korteweg-de Vries (KdV) equation to (2+1) dimensions.…”

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confidence: 99%

“…Derivations of (1) have been given in [4] for those different cases, characterized by different values of s. See also [5,10] for details.…”

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confidence: 99%

“…Derivations of (1) have been given in [4] for those different cases, characterized by different values of s. See also [5,10] for details. [4] presents several approximate analytic solutions, subject to either periodic or Dirichlet boundary conditions in the direction transverse to the propagation (i. e., the y direction), which are the travelling solitary waves having damped oscillations and propagating in a channel (along the x-axis).…”

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confidence: 99%

“…The instability treatment [5] indicates that travelling periodic waves subject to xhomogeneous y-axis perturbations, also analytic but approximate, are found to decay into a sequence of parallel wave guides or channels [5], each of which represents a wave propagating along the x axis but lo-calized in the y direction. The details on the types of wave propagation, including the stability of solutions and the presence of solitary wave solutions, can be seen in [4,5,10]. For s = −1, [9] proposes some possibly observable effects for soliton-like liquid wave propagation in the presence of surface tension and [6] reports the instability and collapse of waveguides on the water surface under the ice cover.…”

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confidence: 99%

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Computerized Symbolic Computation on a Sixth-order Model for Liquid Waves in the Presence of Surface Tension or a Floating Ice

Tian

Gao

2004

Zeitschrift Für Naturforschung A

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Computerized symbolic computation reflects the rapid expansion of computer sciences in various fields of science and engineering, while the studies on the liquid surfaces for rivers, oceans, aviation kerosene, liquid propellant for rockets, etc., are of current interest. In the presence of surface tension or sea ice, and with symbolic computation, the Hȃrȃgus-Courcelle-Il'ichev model for surface liquid waves is hereby investigated. Several similarity reductions are presented, some of which are explicitly written out as exact analytic solutions having their rational expressions with respect to the dimensionless spatial variables of the model. Investigations on the liquid surfaces are of interest, such as those for rivers, oceans, aviation kerosene and liquid propellant for rockets. In the presence of surface tension or sea ice, the flexural-gravity or water-ice waves are of practical value in such studies as those on the damage to offshore constructions by floating ice sheets, ice growth on structures and stress control for the facilities built upon the ice. Readers interested in those topics are referred to [1 -7]. On the other hand, for the gravity-capillary case, the experimental data and theoretical results are relatively hard to be compared [8,9].Among the intesting ones, the Hȃrȃgus-CourcelleIl'ichev model for the gravity-capillary and flexuralgravity waves,has recently been proposed [4,5,10], which is (2+1)-dimensional and 6th-ordered, where x, y and t are dimensionless spatial and temporal variables and u is a dimensionless surface deviation. Equation (1) generalizes the Kadomtsev-Petviashvili (KP) equation to the presence of higher order dispersive effects (caused either by sea ice or to surface tension), and the fifth order 0932-0784 / 04 / 1200-0997 $ 06.00 c 2004 Verlag der Zeitschrift für Naturforschung, Tübingen · http://znaturforsch.com Korteweg-de Vries (KdV) equation to (2+1) dimensions. For the standard KP equation, plane soliton solutions may be unstableor stable depending on the sign of the dispersion (hereby, sign of s). Also, [11] reports that the higher order dispersive terms are related to the radiation of the KdV solitons. Advanced topics on the KP equation can be found, e. g., in [12 -17], while on the 5th-order KdV equation, e. g., in [4, 18 -20].With respect to (1) for dimensionless bond numbers b > 1 3 , and for ice plates with large initial tensions, s = −1; otherwise s = 1. Derivations of (1) have been given in [4] for those different cases, characterized by different values of s. See also [5,10] for details.[4] presents several approximate analytic solutions, subject to either periodic or Dirichlet boundary conditions in the direction transverse to the propagation (i. e., the y direction), which are the travelling solitary waves having damped oscillations and propagating in a channel (along the x-axis). The instability treatment [5] indicates that travelling periodic waves subject to xhomogeneous y-axis perturbations, also analytic but approximate, are found to decay into a sequenc...

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confidence: 99%

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confidence: 99%

“…Derivations of (1) have been given in [4] for those different cases, characterized by different values of s. See also [5,10] for details.…”

mentioning

confidence: 99%

mentioning

confidence: 99%