scholarly journals Hamiltonian formulation for the description of interfacial solitary waves

1998 ◽  
Vol 5 (1) ◽  
pp. 3-12 ◽  
Author(s):  
R. Grimshaw ◽  
S. R. Pudjaprasetya
Keyword(s):  
Solitary Waves ◽  
Variable Coefficient ◽  
Hamiltonian Structure ◽  
Vries Equation ◽  
Hamiltonian Formulation ◽  
Constant Density ◽  
Two Fluids ◽  
De Vries ◽  
Lower Fluid ◽  
New Equation

Abstract. We consider solitary waves propagating on the interface between two fluids, each of constant density, for the case when the upper fluid is bounded above by a rigid horizontal plane, but the lower fluid has a variable depth. It is well-known that in this situation, the solitary waves can be described by a variable-coefficient Korteweg-de Vries equation. Here we reconsider the derivation of this equation and present a formulation which preserves the Hamiltonian structure of the underlying system. The result is a new variable-coefficient Korteweg-de Vries equation, which conserves energy to a higher order than the more conventional well-known equation. The new equation is used to describe the transformation of an interfacial solitary wave which propagates into a region of decreasing depth.

Slowly varying solitary waves. I. Korteweg-de Vries equation

1979 ◽  
Vol 368 (1734) ◽  
pp. 359-375 ◽  
Keyword(s):  
Exact Solution ◽  
Solitary Wave ◽  
Asymptotic Solution ◽  
Solitary Waves ◽  
Variable Coefficient ◽  
Vries Equation ◽  
Multiple Scale ◽  
Perturbation Parameter ◽  
Scale Method ◽  
De Vries

The slowly varying solitary wave is constructed as an asymptotic solution of the variable coefficient Korteweg-de Vries equation. A multiple scale method is used to determine the amplitude and phase of the wave to the second order in the perturbation parameter. The structure ahead and behind the solitary wave is also determined, and the results are interpreted by using conservation laws. Outer expansions are introduced to remove non-uniformities in the expansion. Finally, when the coefficients satisfy a certain constraint, an exact solution is constructed.

Weakly nonlinear internal wave fronts trapped in contractions

2000 ◽  
Vol 415 ◽  
pp. 323-345 ◽  
Author(s):  
S. R. CLARKE ◽  
R. H. J. GRIMSHAW
Keyword(s):  
Internal Wave ◽  
Solitary Waves ◽  
Variable Coefficient ◽  
Vries Equation ◽  
Wave Fronts ◽  
Nonlinear Internal Wave ◽  
Weakly Nonlinear ◽  
Virtual Control ◽  
De Vries

The propagation of weakly nonlinear, long internal wave fronts in a contraction is considered in the transcritical limit as a model for the establishment of virtual controls. It is argued that the appropriate equation to describe this process is a variable coefficient Korteweg–de Vries equation. The solutions of this equation are then considered for compressive and rarefaction fronts. Rarefaction fronts exhibit both normal and virtual control solutions. However, the interaction of compressive fronts with contractions is intrinsically unsteady. Here the dynamics take two forms, interactions with the bulk of the front and interactions with individual solitary waves separating off from a front trapped downstream of the contraction.

scholarly journals Painleve analysis for a forced Korteveg-de Vries equation arisen in fluid dynamics of internal solitary waves

2015 ◽  
Vol 19 (4) ◽  
pp. 1223-1226 ◽  
Author(s):  
Sheng Zhang ◽  
Mei-Tong Chen ◽  
Wei-Yi Qian
Keyword(s):  
Fluid Dynamics ◽  
Solitary Waves ◽  
Variable Coefficient ◽  
Vries Equation ◽  
Internal Solitary Waves ◽  
Painlevé Analysis ◽  
New Exact Solutions ◽  
De Vries ◽  
Painleve Analysis ◽  
Truncated Painlevé Expansion

In this paper, Painleve analysis is used to test the Painleve integrability of a forced variable-coefficient extended Korteveg-de Vries equation which can describe the weakly-non-linear long internal solitary waves in the fluid with continuous stratification on density. The obtained results show that the equation is integrable under certain conditions. By virtue of the truncated Painleve expansion, a pair of new exact solutions to the equation is obtained.

Simulation of the Transformation of Internal Solitary Waves on Oceanic Shelves

2004 ◽  
Vol 34 (12) ◽  
pp. 2774-2791 ◽  
Author(s):  
Roger Grimshaw ◽  
Efim Pelinovsky ◽  
Tatiana Talipova ◽  
Audrey Kurkin
Keyword(s):  
Solitary Wave ◽  
Solitary Waves ◽  
Variable Coefficient ◽  
Asymptotic Theory ◽  
Vries Equation ◽  
Internal Solitary Waves ◽  
Variable Depth ◽  
North West ◽  
Wave Parameters ◽  
De Vries

Abstract Internal solitary waves transform as they propagate shoreward over the continental shelf into the coastal zone, from a combination of the horizontal variability of the oceanic hydrology (density and current stratification) and the variable depth. If this background environment varies sufficiently slowly in comparison with an individual solitary wave, then that wave possesses a soliton-like form with varying amplitude and phase. This stage is studied in detail in the framework of the variable-coefficient extended Korteweg–de Vries equation where the variation of the solitary wave parameters can be described analytically through an asymptotic description as a slowly varying solitary wave. Direct numerical simulation of the variable-coefficient extended Korteweg–de Vries equation is performed for several oceanic shelves (North West shelf of Australia, Malin shelf edge, and Arctic shelf) to demonstrate the applicability of the asymptotic theory. It is shown that the solitary wave may maintain its soliton-like form for large distances (up to 100 km), and this fact helps to explain why internal solitons are widely observed in the world's oceans. In some cases the background stratification contains critical points (where the coefficients of the nonlinear terms in the extended Korteweg–de Vries equation change sign), or does not vary sufficiently slowly; in such cases the solitary wave deforms into a group of secondary waves. This stage is studied numerically.

scholarly journals Modelling of Polarity Change in a Nonlinear Internal Wave Train in Laoshan Bay

2016 ◽  
Vol 46 (3) ◽  
pp. 965-974 ◽  
Author(s):  
Roger Grimshaw ◽  
Caixia Wang ◽  
Lan Li
Keyword(s):  
Critical Point ◽  
Solitary Waves ◽  
Variable Coefficient ◽  
Nonlinear Term ◽  
Vries Equation ◽  
Wave Train ◽  
Periodic Wave ◽  
De Vries ◽  
Theoretical Predictions ◽  
Polarity Change

AbstractThere are now several observations of internal solitary waves passing through a critical point where the coefficient of the quadratic nonlinear term in the variable coefficient Korteweg–de Vries equation changes sign, typically from negative to positive as the wave propagates shoreward. This causes a solitary wave of depression to transform into a train of solitary waves of elevation riding on a negative pedestal. However, recently a polarity change of a different kind was observed in Laoshan Bay, China, where a periodic wave train of elevation waves converted to a periodic wave train of depression waves as the thermocline rose on a rising tide. This paper describes the application of a newly developed theory for this phenomenon. The theory is based on the variable coefficient Korteweg–de Vries equation for the case when the coefficient of the quadratic nonlinear term undergoes a change of sign and predicts that a periodic wave train will pass through this critical point as a linear wave, where a phase change occurs that induces a change in the polarity of the wave, as observed. A two-layer model of the density stratification and background current shear is developed to make the theoretical predictions specific and quantitative. Some numerical simulations of the variable coefficient Korteweg–de Vries equation, and also the extended variable coefficient Korteweg–de Vries equation, are reported that confirm the theoretical predictions and are in good agreement with the observations.

Stability and instability of some nonlinear dispersive solitary waves in higher dimension

1996 ◽  
Vol 126 (1) ◽  
pp. 89-112 ◽  
Author(s):  
Anne de Bouard
Keyword(s):  
Solitary Waves ◽  
Acoustic Waves ◽  
Vries Equation ◽  
Three Dimensional ◽  
Higher Dimension ◽  
Ion Acoustic Waves ◽  
Stability And Instability ◽  
Ion Acoustic ◽  
De Vries ◽  
The Stability

We study the stability of positive radially symmetric solitary waves for a three dimensional generalisation of the Korteweg de Vries equation, which describes nonlinear ion-acoustic waves in a magnetised plasma, and for a generalisation in dimension two of the Benjamin–Bona–Mahony equation.

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