scholarly journals HIGHER ORDER KORTEWEG-DE VRIES MODELS FOR INTERNAL WAVES

2003 ◽  
Vol 2 (2) ◽  
pp. 15
Author(s):  
J. JAHARUDDIN
Keyword(s):  
Free Surface ◽  
Evolution Equation ◽  
Linear Theory ◽  
Internal Waves ◽  
Solvability Condition ◽  
Vries Equation ◽  
Asymptotic Methods ◽  
Higher Order ◽  
De Vries ◽  
Order Extension

By using asymptotic methods, evolution equation is derived for the internal waves in density stratified fluid. This evo- lution equation arise as a solvability condition. A higher-order extension of the familiar Korteweg-de Vries equation is produced for internal waves in a density stratified flow with a free surface. All coefficients of this extended Korteweg-de Vries equation are expressed via integrals of the modal function for the linear theory of long internal waves.

scholarly journals Higher-order Korteweg-de Vries models for internal solitary waves in a stratified shear flow with a free surface

2002 ◽  
Vol 9 (3/4) ◽  
pp. 221-235 ◽  
Author(s):  
R. Grimshaw ◽  
E. Pelinovsky ◽  
O. Poloukhina
Keyword(s):  
Free Surface ◽  
Shear Flow ◽  
Solitary Waves ◽  
Vries Equation ◽  
Wave Theory ◽  
Higher Order ◽  
Internal Solitary Waves ◽  
Long Wave ◽  
De Vries ◽  
Stratified Shear Flow

Abstract. A higher-order extension of the familiar Korteweg-de Vries equation is derived for internal solitary waves in a density- and current-stratified shear flow with a free surface. All coefficients of this extended Korteweg-de Vries equation are expressed in terms of integrals of the modal function for the linear long-wave theory. An illustrative example of a two-layer shear flow is considered, for which we discuss the parameter dependence of the coefficients in the extended Korteweg-de Vries equation.

Gauge transformation and the higher order Korteweg–de Vries equation

1988 ◽  
Vol 29 (2) ◽  
pp. 308-314 ◽  
Author(s):  
Yu‐kun Zheng ◽  
W. L. Chan
Keyword(s):  
Gauge Transformation ◽  
Vries Equation ◽  
Higher Order ◽  
De Vries

Noncommutative coupled complex modified Korteweg–de Vries equation: Darboux and binary Darboux transformations

2019 ◽  
Vol 34 (07n08) ◽  
pp. 1950054
Author(s):  
H. Wajahat A. Riaz
Keyword(s):  
Evolution Equations ◽  
Nonlinear Evolution ◽  
Vries Equation ◽  
Soliton Solutions ◽  
Higher Order ◽  
Nonlinear Evolution Equations ◽  
Order Effects ◽  
Darboux Transformations ◽  
De Vries ◽  
Higher Order Effects

Higher-order nonlinear evolution equations are important for describing the wave propagation of second- and higher-order number of fields in optical fiber systems with higher-order effects. One of these equations is the coupled complex modified Korteweg–de Vries (ccmKdV) equation. In this paper, we study noncommutative (nc) generalization of ccmKdV equation. We present Darboux and binary Darboux transformations (DTs) for the nc-ccmKdV equation and then construct its Quasi-Grammian solutions. Further, single and double-hump soliton solutions of first- and second-order are given in commutative settings.

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.

Transcritical flow over obstacles and holes: forced Korteweg–de Vries framework

2019 ◽  
Vol 881 ◽  
pp. 660-678 ◽  
Author(s):  
Roger H. J. Grimshaw ◽  
Montri Maleewong
Keyword(s):  
Free Surface ◽  
Solitary Wave ◽  
Vries Equation ◽  
Solitary Wave Solution ◽  
Free Surface Flow ◽  
Surface Flow ◽  
Main Concern ◽  
Oncoming Flow ◽  
Transcritical Flow ◽  
De Vries

This paper extends a previous study of free-surface flow over two localised obstacles using the framework of the forced Korteweg–de Vries equation, to an analogous study of flow over two localised holes, or a combination of an obstacle and a hole. Importantly the terminology obstacle or hole can be reversed for a stratified fluid and refers more precisely to the relative polarity of the forcing and the solitary wave solution of the unforced Korteweg–de Vries equation. As in the previous study, our main concern is with the transcritical regime when the oncoming flow has a Froude number close to unity. In the transcritical regime at early times, undular bores are produced upstream and downstream of each forcing site. We then describe the interaction of these undular bores between the forcing sites, and the outcome at very large times.

Double-soliton and conservation law structures for a higher-order type of Korteweg-de Vries equation

2015 ◽  
Vol 28 (4) ◽  
pp. 633-638
Author(s):  
C. T. Lee ◽  
C. C. Lee ◽  
M. L. Liu
Keyword(s):  
Conservation Law ◽  
Vries Equation ◽  
Order Type ◽  
Higher Order ◽  
De Vries
Sign in / Sign up

Export Citation Format

Share Document