Three-dimensional solitary waves in the presence of additional surface effects

Hărăguş-Courcelle, Mariana; Ильичев, А. Т.

doi:10.1016/s0997-7546(98)80023-x

Cited by 41 publications

(30 citation statements)

References 16 publications

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“…which are associated with continuous symmetries, namely the invariance of Hamilton's equations under translations in z, x and Φ. Hamilton's equations also inherit the discrete symmetries of the hydrodynamic problem (6)- (9). They are invariant under the reflection R : X s → X s given by…”

Section: Lemma 1 Let T Be a Non-negative Real Number (I) The Mappingmentioning

confidence: 99%

“…Introducing Cartesian coordinates (x, y, z) so that y points vertically upward and x points in the direction of the travelling wave, one finds that the three-dimensional hydrodynamic problem involves two independent unbounded horizontal spatial coordinates x − ct and z, where c denotes the speed of the wave. Motivated by the arguments used for a model equation by HaragusCourcelle and Illichev [9], Groves and Mielke formulated the hydrodynamic equations as a dynamical system in which x − ct is the time-like variable and examined wave motions which are periodic in z. This approach represents a natural step from two to three dimensions since it includes all two-dimensional travelling waves as special cases.…”

Section: Introductionmentioning

confidence: 99%

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An existence theory for three-dimensional periodic travelling gravity–capillary water waves with bounded transverse profiles

Groves

2001

Physica D: Nonlinear Phenomena

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Additional Information:• This is a pre-print. This pre-print appeared in the journal, Physica D An existence theory for three-dimensional periodic travelling gravity-capillary water waves with bounded transverse profiles M. D. Groves Department of Mathematical Sciences, Loughborough University, Loughborough, LE11 3TU, UKAbstract: This article presents a rigorous existence theory for threedimensional gravity-capillary water waves which are uniformly translating and periodic in one horizontal spatial direction x and have a nontrivial transverse profile in the other z. The hydrodynamic equations are formulated as an infinite-dimensional Hamiltonian system in which z is the time-like variable, and a centre-manifold reduction technique is applied to demonstrate that the problem is locally equivalent to a finite-dimensional Hamiltonian system of ordinary differential equations.A family of straight lines C 1 , C 2 , . . . in Bond number-Froude number parameter space is identified which are associated with codimension-one bifurcation phenomena: at each point on one of these lines two real eigenvalues of the linear problem become purely imaginary by passing through zero. There are also codimension-two points: the line C k intersects each of the lines C k+1 , C k+2 , . . . in precisely one point.General statements concerning the existence of waves which are periodic or quasiperiodic in z are made by applying standard tools in Hamiltoniansystems theory to the reduced equations. Moreover, a critical curve in parameter space is found at which a wave with a spatially localised and exponentially decaying transverse profile bifurcates from zero. This curve is piecewise linear: it contains one line segment from each of C 1 , C

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Section: Lemma 1 Let T Be a Non-negative Real Number (I) The Mappingmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

An existence theory for three-dimensional periodic travelling gravity–capillary water waves with bounded transverse profiles

Groves

2001

Physica D: Nonlinear Phenomena

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“…The former reference considers waves which are periodic in the transverse spatial direction z and uses the longitudinal variable x as the time-like variable, while the latter considers waves which are periodic in the x-direction and uses z as the time-like variable. Both of these choices, which were motivated by similar studies of model equations by respectively Haragus-Courcelle & Ilichev [10] and Haragus-Courcelle & Pego [11], represent natural steps from two to three dimensions: the former includes all two-dimensional travelling waves as special cases, while the latter facilitates a discussion of the 'dimension-breaking' phenomenon in which two dimensional waves spontaneously lose their spatial inhomogeneity in the z-direction (see Groves, Haragus & Sun [8]). …”

Section: Hamiltonian Spatial Dynamics and Travelling Water Wavesmentioning

confidence: 99%

A plethora of three-dimensional periodic travelling gravity-capillary water waves with multipulse transverse profiles

Groves

Sandstede

2004

J Nonlinear Sci

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“…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.…”

mentioning

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%

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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Three-dimensional solitary waves in the presence of additional surface effects

Cited by 41 publications

References 16 publications

An existence theory for three-dimensional periodic travelling gravity–capillary water waves with bounded transverse profiles

An existence theory for three-dimensional periodic travelling gravity–capillary water waves with bounded transverse profiles

A plethora of three-dimensional periodic travelling gravity-capillary water waves with multipulse transverse profiles

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

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