Depression and elevation tsunami waves in the framework of the Korteweg–de Vries equation

Although tsunamis in the deep ocean are very long waves of quite small amplitudes, as they propagate shorewards into shallow water, nonlinearity becomes important and the structure of the leading waves depends on the polarity of the incident wave from the deep ocean. In this paper, we use a variable-coefficient Korteweg–de Vries equation to examine this issue, for an initial wave which is either elevation, or depression, or a combination of each. We show that the leading waves can be described by a reduction of the Whitham modulation theory to a solitary wave train. We find that for an initial elevation, the leading waves are elevation solitary waves with an amplitude which varies inversely with the depth, with a pre-factor which is twice the maximum amplitude in the initial wave. By contrast, for an initial depression, the leading wave is a depression rarefaction wave, followed by a solitary wave train whose maximum amplitude of the leading wave is determined by the square root of the mass in the initial wave

Generation of solitary waves by transcritical flow over a step

It is well-known that transcritical flow over a localized obstacle generates upstream and downstream nonlinear wavetrains. The flow has been successfully modelled in the framework of the forced Korteweg - de Vries equation, where numerical and asymptotic analytical solutions have shown that the upstream and downstream nonlinear wavetrains have the structure of unsteady undular bores, connected by a locally steady solution over the obstacle, which is elevated on the upstream side and depressed on the downstream side. Inthispaper we consider the analogous transcritical flow over a step, primarily in the context of water waves. We use numerical and asymptotic analytical solutions of the forced Korteweg - de Vries equation, together with numerical solutions of the full Eulerequations, to demonstrate that a positive step generates only an upstream-propagating undular bore, and a negative step generates only a downstream-propagating undular bore. © 2007 Cambridge University Press.published_or_final_versio

Interactions of breathers and solitons in the extended Korteweg-de Vries equation

The extended Korteweg-de Vries model governs the evolution of weakly dispersive waves under the combined influence of quadratic and cubic nonlinearities, and is relevant to finite-amplitude wave motions in the atmosphere and the ocean. Analytic expressions for a multi-soliton are obtained by the Hirota bilinear method, and are shown to agree with those for isolated solitary waves or breathers obtained earlier in the literature. In particular, the interaction of a breather and a soliton can now be studied. Both the soliton and the breather retain their identities after interaction except for some phase shifts. Detailed examination of the interaction process shows that the profile of the breather will depend critically on the polarity of the colliding soliton. © 2005 Elsevier B.V. All rights reserved.postprin

Changing forms and sudden smooth transitions of tsunami waves

In some tsunami waves travelling over the ocean, such as the one approaching the eastern coast of Japan in 2011, the sea surface of the ocean is depressed by a small metre-scale displacement over a multi-kilometre horizontal length scale, lying in front of a positive elevation of comparable magnitude and length, which together constitute a down-up wave. Shallow water theory shows that the latter travels faster than the former, leading to an interaction, whose description is the issue addressed in this paper, using model equations of the Korteweg–de Vries type. First, we re-examine the undular bore solutions of the Korteweg–de Vries equation which describe how an initial depression wave deforms into a depression rarefaction wave followed by an undular bore of large elevation waves riding on this depression. Then we develop a new extended Korteweg–de Vries equation some of whose solutions can be used to describe the interaction of an elevation wave chasing a depression wave. These show that the two waves coincide at a given position and time producing a maximum elevation. Typically this amplitude is larger than the initial displacement magnitude by a factor which can be as large as two, which may explain anomalous elevations of tsunamis at particular positions along their trajectories. It is physically significant that for these small amplitude waves, no wave breaking occurs and there is no excess dissipation. Then, following the transition, the elevation wave moves ahead of the depression wave and the distance between them increases either linearly or logarithmically with time. The implications for how these down-up tsunami waves reach the shoreline are considered.postprin

A coupled 'AB' system: Rogue waves and modulation instabilities

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Interactions of breathers and solitons of the extended Korteweg - de Vries equation

Session HL: Waves II, Abstract no. HL.00002published_or_final_versio

Experimental study of the effect of rotation on large amplitude internal waves

Nonlinear internal waves are commonly observed in the coastal ocean. In the weakly nonlinear long wave régime, they are often modeled by the Korteweg-de Vries equation, which predicts that the long-time outcome of generic localised initial conditions is a train of internal solitary waves. However, when the effect of background rotation is taken into account, it is known from several theoretical and numerical studies that the formation of solitary waves is inhibited, and instead nonlinear wave packets form. In this paper, we report the results from a laboratory experiment at the LEGI-Coriolis Laboratory which describes this process

Rogue wave modes for the long wave-short wave resonance model

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D'Alembert‐type solution of the Cauchy problem for the Boussinesq‐Klein‐Gordon equation

In this paper, we construct a weakly‐nonlinear d'Alembert‐type solution of the Cauchy problem for the Boussinesq‐Klein‐Gordon (BKG) equation. Similarly to our earlier work based on the use of spatial Fourier series, we consider the problem in the class of periodic functions on an interval of finite length (including the case of localized solutions on a large interval), and work with the nonlinear partial differential equation with variable coefficients describing the deviation from the oscillating mean value. Unlike our earlier paper, here we develop a novel multiple‐scales procedure involving fast characteristic variables and two slow time scales and averaging with respect to the spatial variable at a constant value of one or another characteristic variable, which allows us to construct an explicit and compact d'Alembert‐type solution of the nonlinear problem in terms of solutions of two Ostrovsky equations emerging at the leading order and describing the right‐ and left‐propagating waves. Validity of the constructed solution in the case when only the first initial condition for the BKG equation may have nonzero mean value follows from our earlier results, and is illustrated numerically for a number of instructive examples, both for periodic solutions on a finite interval, and localized solutions on a large interval. We also outline an extension of the procedure to the general case, when both initial conditions may have nonzero mean values. Importantly, in all cases, the initial conditions for the leading‐order Ostrovsky equations by construction have zero mean, while initial conditions for the BKG equation may have nonzero mean values