Solitary internal waves with oscillatory tails

Solitary internal waves in a density-stratified fluid of shallow depth are considered. According to the classical weakly nonlinear long-wave theory, the propagation of each long-wave mode is governed by the Korteweg-de Vries equation to leading order, and locally confined solitary waves with a 'sech2' profile are possible. Using a singular-perturbation procedure, it is shown that, in general, solitary waves of mode n>1 actually develop oscillatory tails of infinite extent, consisting of lower-mode short waves. The amplitude of these tails is exponentially small with respect to an amplitude parameter, and lies beyond all orders of the usual long-wave expansion. To illustrate the theory, two special cases of stratification are discussed in detail, and the amplitude of the oscillations at the solitary-wave tails is determined explicitly. The theoretical predictions are supported by experimental observations.

[1]  M. Longuet-Higgins,et al.  On the speed and profile of steep solitary waves , 1976, Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences.

[2]  T. Chan,et al.  Large Amplitude Internal Waves of Permanent Form , 1982 .

[3]  Gareth P. Williams,et al.  The Stability and Genesis of Rossby Vortices , 1988 .

[4]  John K. Hunter,et al.  Existence of perturbed solitary wave solutions to a model equation for water waves , 1988 .

[5]  J. Toland,et al.  Solitary waves with surface tension I: Trajectories homoclinic to periodic orbits in four dimensions , 1992 .

[6]  D. J. Benney.  Long Non‐Linear Waves in Fluid Flows , 1966 .

[7]  Harvey Segur,et al.  Asymptotics beyond all orders in a model of crystal growth , 1991 .

[8]  A. Acrivos,et al.  Solitary internal waves in deep water , 1967, Journal of Fluid Mechanics.

[9]  S. Sun.  Existence of a generalized solitary wave solution for water with positive bond number less than 13 , 1991 .

[10]  John P. Boyd,et al.  Weakly non-local solutions for capillary-gravity waves: fifth-degree Korteweg-de Vries equation , 1991 .

[11]  A. P. Hooper,et al.  The non-existence of a certain class of travelling wave solutions of the Kuramoto-Sivashinsky equation , 1991 .

[12]  D. Farmer,et al.  Tidal interaction of stratified flow with a sill in Knight Inlet , 1980 .

[13]  J. K. Hunter,et al.  Solitary and periodic gravity—capillary waves of finite amplitude , 1983, Journal of Fluid Mechanics.

[14]  S. Orszag,et al.  Advanced Mathematical Methods For Scientists And Engineers , 1979 .

[15]  Alfred Ramani,et al.  Structural stability of the Korteweg-De Vries solitons under a singular perturbation , 1988 .

[16]  J. F. Toland,et al.  On solitary water-waves of finite amplitude , 1981 .

[17]  M. Kruskal,et al.  Nonexistence of small-amplitude breather solutions in phi4 theory. , 1987, Physical review letters.

[18]  J. Thomas Beale,et al.  Exact solitary water waves with capillary ripples at infinity , 1991 .

[19]  A Computational Method for Solitary Internal Waves in a Continuously Stratified Fluid , 1991 .

[20]  J. Vanden-Broeck.  Elevation solitary waves with surface tension , 1991 .

[21]  J. Vanden-Broeck,et al.  Long periodic internal waves , 1992 .

[22]  Michael Selwyn Longuet-Higgins,et al.  On the mass, momentum, energy and circulation of a solitary wave. II , 1974, Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences.