Cnoidal Wave Trains and Solitary Waves in a Dissipation-Modified Korteweg—de Vries Equation

KdV ’95 ◽  
1995 ◽  
pp. 457-475
Author(s):  
A. Ye. Rednikov ◽  
M. G. Velarde ◽  
Yu. S. Ryazantsev ◽  
A. A. Nepomnyashchy ◽  
V. N. Kurdyumov
Keyword(s):  
Solitary Waves ◽  
Vries Equation ◽  
Cnoidal Wave ◽  
De Vries ◽  
Wave Trains

Cnoidal wave trains and solitary waves in a dissipation-modified Korteweg-de Vries equation

1995 ◽  
Vol 39 (1-3) ◽  
pp. 457-475 ◽  
Author(s):  
A. Ye. Rednikov ◽  
M. G. Velarde ◽  
Yu. S. Ryazantsev ◽  
A. A. Nepomnyashchy ◽  
V. N. Kurdyumov
Keyword(s):  
Solitary Waves ◽  
Vries Equation ◽  
Cnoidal Wave ◽  
De Vries ◽  
Wave Trains

SOLITARY WAVES, SOLITON BOUND STATES AND CHAOS IN A DISSIPATIVE KORTEWEG-DE VRIES EQUATION

1994 ◽  
Vol 04 (05) ◽  
pp. 1135-1146 ◽  
Author(s):  
VLADIMIR I. NEKORKIN ◽  
MANUEL G. VELARDE
Keyword(s):  
Internal Waves ◽  
Solitary Waves ◽  
Liquid Layer ◽  
Bound States ◽  
Vries Equation ◽  
Stratified Fluid ◽  
Localized Structures ◽  
Fluid Layers ◽  
De Vries ◽  
Wave Trains

Propagating dissipative (localized) structures like solitary waves, pulses or “solitons,” “bound solitons,” and “chaotic” wave trains are shown to be solutions of a dissipation-modified Korteweg-de Vries equation that in particular appears in Marangoni-Bénard convection when a liquid layer is heated from the air side and in the description of internal waves in sheared, stably stratified fluid layers.

FURTHER RESULTS ON THE EVOLUTION OF SOLITARY WAVES AND THEIR BOUND STATES OF A DISSIPATIVE KORTEWEG-DE VRIES EQUATION

1995 ◽  
Vol 05 (03) ◽  
pp. 831-839 ◽  
Author(s):  
MANUEL G. VELARDE ◽  
VLADIMIR I. NEKORKIN ◽  
ANDREY G. MAKSIMOV
Keyword(s):  
Solitary Waves ◽  
Bound States ◽  
Marangoni Convection ◽  
Vries Equation ◽  
Dissipative Structures ◽  
Long Wavelength ◽  
Numerical Evidence ◽  
De Vries ◽  
Wave Trains

Numerical evidence is provided of (long lasting) solitary waves (traveling localized dissipative structures) and their bound states as well as periodic and “chaotic”, erratic wave trains of a dissipation-modified Korteweg-de Vries equation originally derived to account for long wavelength oscillatory Bénard-Marangoni convection.

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.

Solitary waves with damped oscillatory tails: an analysis of the fifth-order Korteweg-de Vries equation

1994 ◽  
Vol 77 (4) ◽  
pp. 473-485 ◽  
Author(s):  
R. Grimshaw ◽  
B. Malomed ◽  
E. Benilov
Keyword(s):  
Solitary Waves ◽  
Vries Equation ◽  
De Vries ◽  
Fifth Order

scholarly journals Computational and Numerical Solutions for 2+1-Dimensional Integrable Schwarz–Korteweg–de Vries Equation with Miura Transform

Complexity ◽  
2020 ◽  
Vol 2020 ◽  
pp. 1-13
Author(s):  
Raghda A. M. Attia ◽  
S. H. Alfalqi ◽  
J. F. Alzaidi ◽  
Mostafa M. A. Khater ◽  
Dianchen Lu
Keyword(s):  
Solitary Waves ◽  
Decomposition Method ◽  
Numerical Solutions ◽  
Vries Equation ◽  
Adomian Decomposition Method ◽  
Simplest Equation Method ◽  
Adomian Decomposition ◽  
Tanh Method ◽  
De Vries ◽  
Parameter Values

This paper investigates the analytical, semianalytical, and numerical solutions of the 2+1–dimensional integrable Schwarz–Korteweg–de Vries (SKdV) equation. The extended simplest equation method, the sech-tanh method, the Adomian decomposition method, and cubic spline scheme are employed to obtain distinct formulas of solitary waves that are employed to calculate the initial and boundary conditions. Consequently, the numerical solutions of this model can be investigated. Moreover, their stability properties are also analyzed. The solutions obtained by means of these techniques are compared to unravel relations between them and their characteristics illustrated under the suitable choice of the parameter values.

The effect of weak shear on finite-amplitude internal solitary waves

1999 ◽  
Vol 395 ◽  
pp. 125-159 ◽  
Author(s):  
S. R. CLARKE ◽  
R. H. J. GRIMSHAW
Keyword(s):  
Evolution Equation ◽  
Solitary Waves ◽  
Vries Equation ◽  
Physical Space ◽  
Finite Amplitude ◽  
Initial Condition ◽  
Weak Current ◽  
Inflection Points ◽  
Wide Range ◽  
De Vries

A finite-amplitude long-wave equation is derived to describe the effect of weak current shear on internal waves in a uniformly stratified fluid. This effect is manifested through the introduction of a nonlinear term into the amplitude evolution equation, representing a projection of the shear from physical space to amplitude space. For steadily propagating waves the evolution equation reduces to the steady version of the generalized Korteweg–de Vries equation. An analysis of this equation is presented for a wide range of possible shear profiles. The type of waves that occur is found to depend on the number and position of the inflection points of the representation of the shear profile in amplitude space. Up to three possible inflection points for this function are considered, resulting in solitary waves and kinks (dispersionless bores) which can have up to three characteristic lengthscales. The stability of these waves is generally found to decrease as the complexity of the waves increases. These solutions suggest that kinks and solitary waves with multiple lengthscales are only possible for shear profiles (in physical space) with a turning point, while instability is only possible if the shear profile has an inflection point. The unsteady evolution of a periodic initial condition is considered and again the solution is found to depend on the inflection points of the amplitude representation of the shear profile. Two characteristic types of solution occur, the first where the initial condition evolves into a train of rank-ordered solitary waves, analogous to those generated in the framework of the Korteweg–de Vries equation, and the second where two or more kinks connect regions of constant amplitude. The unsteady solutions demonstrate that finite-amplitude effects can act to halt the critical collapse of solitary waves which occurs in the context of the generalized Korteweg–de Vries equation. The two types of solution are then used to qualititatively relate previously reported observations of shock formation on the internal tide propagating onto the Australian North West Shelf to the observed background current shear.

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.

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