The Asymptotic Approach to the Description of Two-Dimensional Symmetric Soliton Patterns

The asymptotic approach is suggested for the description of interacting surface and internal oceanic solitary waves. This approach allows one to describe stationary moving symmetric wave patterns consisting of two plane solitary waves of equal amplitudes moving at an angle to each other. The results obtained within the approximate asymptotic theory are validated by comparison with the exact two-soliton solution of the Kadomtsev–Petviashvili equation (KP2-equation). The suggested approach is equally applicable to a wide class of non-integrable equations too. As an example, the asymptotic theory is applied to the description of wave patterns in the 2D Benjamin–Ono equation describing internal waves in the infinitely deep ocean containing a relatively thin one of the layers.

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Asymptotic expansions for solitary gravity-capillary waves in two and three dimensions

Proceedings of The Royal Society A Mathematical Physical and Engineering Sciences ◽

10.1098/rspa.2009.0112 ◽

2009 ◽

Vol 465 (2109) ◽

pp. 2725-2749 ◽

Cited By ~ 4

Author(s):

M. J. Ablowitz ◽

T. S. Haut

Keyword(s):

Surface Tension ◽

Solitary Wave ◽

Solitary Waves ◽

Asymptotic Series ◽

High Order ◽

Three Dimensions ◽

Two Dimensional ◽

Small Surface ◽

Petviashvili Equation ◽

Dimensional Series

High-order asymptotic series are obtained for two- and three-dimensional gravity-capillary solitary waves. In two dimensions, the first term in the asymptotic series is the well-known sech 2 solution of the Korteweg–de Vries equation; in three dimensions, the first term is the rational lump solution of the Kadomtsev–Petviashvili equation I. The two-dimensional series is used (with nine terms included) to investigate how small surface tension affects the height and energy of large-amplitude waves and waves close to the solitary version of Stokes’ extreme wave. In particular, for small surface tension, the solitary wave with the maximum energy is obtained. For large surface tension, the two-dimensional series is also used to study the energy of depression solitary waves. Energy considerations suggest that, for large enough surface tension, there are solitary waves that can get close to the fluid bottom. In three dimensions, analytic solutions for the high-order perturbation terms are computed numerically, and the resulting asymptotic series (to three terms) is used to obtain the speed versus maximum amplitude curve for solitary waves subject to sufficiently large surface tension. Finally, the above asymptotic method is applied to the Benney–Luke (BL) equation, and the resulting asymptotic series (to three terms) is verified to agree with the solitary-wave solution of the BL equation.

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Two-dimensional Interaction of Solitary Waves in a Modified Kadomtsev–Petviashvili Equation

Journal of the Physical Society of Japan ◽

10.1143/jpsj.73.3034 ◽

2004 ◽

Vol 73 (11) ◽

pp. 3034-3043 ◽

Cited By ~ 13

Author(s):

Hidekazu Tsuji ◽

Masayuki Oikawa

Keyword(s):

Solitary Waves ◽

Two Dimensional ◽

Dimensional Interaction ◽

Petviashvili Equation

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The Propagation of Internal Solitary Waves over Variable Topography in a Horizontally Two-Dimensional Framework

Journal of Physical Oceanography ◽

10.1175/jpo-d-17-0154.1 ◽

2018 ◽

Vol 48 (2) ◽

pp. 283-300 ◽

Cited By ~ 7

Author(s):

Chunxin Yuan ◽

Roger Grimshaw ◽

Edward Johnson ◽

Xueen Chen

Keyword(s):

Solitary Wave ◽

Wave Field ◽

Solitary Waves ◽

Variable Coefficient ◽

Transverse Direction ◽

Density Stratification ◽

Internal Solitary Waves ◽

Two Dimensional ◽

Petviashvili Equation ◽

Background Density

AbstractThis paper presents a horizontally two-dimensional theory based on a variable-coefficient Kadomtsev–Petviashvili equation, which is developed to investigate oceanic internal solitary waves propagating over variable bathymetry, for general background density stratification and current shear. To illustrate the theory, a typical monthly averaged density stratification is used for the propagation of an internal solitary wave over either a submarine canyon or a submarine plateau. The evolution is essentially determined by two components, nonlinear effects in the main propagation direction and the diffraction modulation effects in the transverse direction. When the initial solitary wave is located in a narrow area, the consequent spreading effects are dominant, resulting in a wave field largely manifested by a significant diminution of the leading waves, together with some trailing shelves of the opposite polarity. On the other hand, if the initial solitary wave is uniform in the transverse direction, then the evolution is more complicated, though it can be explained by an asymptotic theory for a slowly varying solitary wave combined with the generation of trailing shelves needed to satisfy conservation of mass. This theory is used to demonstrate that it is the transverse dependence of the nonlinear coefficient in the Kadomtsev–Petviashvili equation rather than the coefficient of the linear transverse diffraction term that determines how the wave field evolves. The Massachusetts Institute of Technology (MIT) general circulation model is used to provide a comparison with the variable-coefficient Kadomtsev–Petviashvili model, and good qualitative and quantitative agreements are found.

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Blow up and instability of solitary wave solutions to a generalized Kadomtsev–Petviashvili equation and two-dimensional Benjamin–Ono equations

Proceedings of The Royal Society A Mathematical Physical and Engineering Sciences ◽

10.1098/rspa.2007.0013 ◽

2007 ◽

Vol 464 (2089) ◽

pp. 49-64 ◽

Cited By ~ 2

Author(s):

Jianqing Chen ◽

Boling Guo ◽

Yongqian Han

Keyword(s):

Initial Data ◽

Solitary Wave ◽

Solitary Waves ◽

Blow Up ◽

Positive Energy ◽

Two Dimensional ◽

Solitary Wave Solutions ◽

Wave Solutions ◽

Petviashvili Equation ◽

Special Case

Let with p being the ratio of an even to an odd integer. For the generalized Kadomtsev–Petviashvili equation, coupled with Benjamin–Ono equations, in the form it is proved that the solutions blow up in finite time even for those initial data with positive energy. As a by-product, it is proved that for all , the solitary waves are strongly unstable if . This result, even in a special case , improves a previous work by Liu (Liu 2001 Trans. AMS 353 , 191–208) where the instability of solitary waves was proved only in the case of .

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Two-Dimensional Solitary Waves in a Strained Nonlinear Viscoelastic Medium

Acoustical Physics ◽

10.1134/1.29853 ◽

2000 ◽

Vol 46 (1) ◽

pp. 100 ◽

Cited By ~ 1

Author(s):

G. A. Arshinov

Keyword(s):

Solitary Waves ◽

Viscoelastic Medium ◽

Two Dimensional ◽

Nonlinear Viscoelastic ◽

Nonlinear Viscoelastic Medium

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Two-dimensional leaps in near-critical flow over isolated orography

Proceedings of The Royal Society A Mathematical Physical and Engineering Sciences ◽

10.1098/rspa.2005.1530 ◽

2005 ◽

Vol 461 (2064) ◽

pp. 3747-3763 ◽

Cited By ~ 3

Author(s):

E.R Johnson ◽

G.G Vilenski

Keyword(s):

Unsteady Flow ◽

Equations Of Motion ◽

Fold Increase ◽

Flow Speed ◽

Two Dimensional ◽

Subcritical Flow ◽

One Dimensional ◽

Petviashvili Equation ◽

Lee Wave ◽

Supercritical Flows

This paper describes steady two-dimensional disturbances forced on the interface of a two-layer fluid by flow over an isolated obstacle. The oncoming flow speed is close to the linear longwave speed and the layer densities, layer depths and obstacle height are chosen so that the equations of motion reduce to the forced two-dimensional Korteweg–de Vries equation with cubic nonlinearity, i.e. the forced extended Kadomtsev–Petviashvili equation. The distinctive feature noted here is the appearance in the far lee-wave wake behind obstacles in subcritical flow of a ‘table-top’ wave extending almost one-dimensionally for many obstacles widths across the flow. Numerical integrations show that the most important parameter determining whether this wave appears is the departure from criticality, with the wave appearing in slightly subcritical flows but being destroyed by two-dimensional effects behind even quite long ridges in even moderately subcritical flow. The wave appears after the flow has passed through a transition from subcritical to supercritical over the obstacle and its leading and trailing edges resemble dissipationless leaps standing in supercritical flow. Two-dimensional steady supercritical flows are related to one-dimensional unsteady flows with time in the unsteady flow associated with a slow cross-stream variable in the two-dimensional flows. Thus the wide cross-stream extent of the table-top wave appears to derive from the combination of its occurrence in a supercritical region embedded in the subcritical flow and the propagation without change of form of table-top waves in one-dimensional unsteady flow. The table-top wave here is associated with a resonant steepening of the transition above the obstacle and a consequent twelve-fold increase in drag. Remarkably, the table-top wave is generated equally strongly and extends laterally equally as far behind an axisymmetric obstacle as behind a ridge and so leads to subcritical flows differing significantly from linear predictions.

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Comparison principles for free-surface flows with gravity

Journal of Fluid Mechanics ◽

10.1017/s0022112091000770 ◽

1991 ◽

Vol 230 ◽

pp. 231-243 ◽

Cited By ~ 3

Author(s):

Walter Craig ◽

Peter Sternberg

Keyword(s):

Solitary Waves ◽

Finite Depth ◽

Free Surface Flows ◽

Immiscible Fluids ◽

Steady Flows ◽

Two Dimensional ◽

Surface Flows ◽

Ship Hulls ◽

Geometrical Information ◽

Supercritical Flows

This article considers certain two-dimensional, irrotational, steady flows in fluid regions of finite depth and infinite horizontal extent. Geometrical information about these flows and their singularities is obtained, using a variant of a classical comparison principle. The results are applied to three types of problems: (i) supercritical solitary waves carrying planing surfaces or surfboards, (ii) supercritical flows past ship hulls and (iii) supercritical interfacial solitary waves in systems consisting of two immiscible fluids.

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