Weakly non-linear waves in rotating fluids

1970 ◽  
Vol 42 (4) ◽  
pp. 803-822 ◽  
Author(s):  
S. Leibovich
Keyword(s):  
Differential Equation ◽  
Singular Perturbation ◽  
Vortex Breakdown ◽  
Vries Equation ◽  
Tube Wall ◽  
Rotating Fluids ◽  
Integro Differential Equation ◽  
Stationary Flows ◽  
Linear Waves ◽  
De Vries

The Korteweg–de Vries equation is shown to govern formation of solitary and cnoidal waves in rotating fluids confined in tubes. It is proved that the method must fail when the tube wall is moved to infinity, and the failure is corrected by singular perturbation procedures. The Korteweg–de Vries equation must then give way to an integro-differential equation. Also, critical stationary flows in tubes are considered with regard to Benjamin's vortex breakdown theories.

Asymptotic stepwise solutions of the Korteweg--de Vries equation with a singular perturbation and their accuracy

2021 ◽  
Vol 8 (3) ◽  
pp. 410-421
Author(s):  
S. I. Lyashko ◽  
◽  
V. H. Samoilenko ◽  
Yu. I. Samoilenko ◽  
I. V. Gapyak ◽  
...  
Keyword(s):  
Singular Perturbation ◽  
Small Parameter ◽  
Asymptotic Approximation ◽  
Vries Equation ◽  
Approximate Solutions ◽  
Variable Coefficients ◽  
Main Term ◽  
Asymptotic Approximations ◽  
De Vries ◽  
The Given

The paper deals with the Korteweg-de Vries equation with variable coefficients and a small parameter at the highest derivative. The asymptotic step-like solution to the equation is obtained by the non-linear WKB technique. An algorithm of constructing the higher terms of the asymptotic step-like solutions is presented. The theorem on the accuracy of the higher asymptotic approximations is proven. The proposed technique is demonstrated by example of the equation with given variable coefficients. The main term and the first asymptotic approximation of the given example are found, their analysis is done and statement of the approximate solutions accuracy is presented.

scholarly journals Existence of periodic traveling wave solution to the forced generalized nearly concentric Korteweg-de Vries equation

2000 ◽  
Vol 24 (6) ◽  
pp. 371-377 ◽  
Author(s):  
Kenneth L. Jones ◽  
Xiaogui He ◽  
Yunkai Chen
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Integral Equations ◽  
Traveling Wave ◽  
Vries Equation ◽  
Traveling Wave Solutions ◽  
Wave Solutions ◽  
Petviashvili Equation ◽  
De Vries ◽  
Periodic Traveling Wave

This paper is concerned with periodic traveling wave solutions of the forced generalized nearly concentric Korteweg-de Vries equation in the form of(uη+u/(2η)+[f(u)]ξ+uξξξ)ξ+uθθ/η2=h0. The authors first convert this equation into a forced generalized Kadomtsev-Petviashvili equation,(ut+[f(u)]x+uxxx)x+uyy=h0, and then to a nonlinear ordinary differential equation with periodic boundary conditions. An equivalent relationship between the ordinary differential equation and nonlinear integral equations with symmetric kernels is established by using the Green's function method. The integral representations generate compact operators in a Banach space of real-valued continuous functions. The Schauder's fixed point theorem is then used to prove the existence of nonconstant solutions to the integral equations. Therefore, the existence of periodic traveling wave solutions to the forced generalized KP equation, and hence the nearly concentric KdV equation, is proved.

Comments on the use of the Korteweg-de Vries equation in the study of anharmonic lattices

1974 ◽  
Vol 339 (1616) ◽  
pp. 119-126 ◽  
Keyword(s):  
Differential Equation ◽  
Lattice Dynamics ◽  
Continuum Limit ◽  
Vries Equation ◽  
Wave Dispersion ◽  
Order Partial Differential Equation ◽  
Lennard Jones ◽  
Long Wave ◽  
De Vries ◽  
The Continuum

The use of the Korteweg-de Vries equation as the continuum limit of the equations describing the anharmonic motion of atoms in a lattice is examined in the light of the periodic solutions recently constructed by Askar (1973). It is shown that the Korteweg-de Vries equation does not exactly represent the behaviour of the nonlinear lattice in the limit of long waves and that a sixth-order partial differential equation gives a more accurate description of the lattice dynamics. The relative accuracies of two long wave dispersion relations derived from the Korteweg-de Vries equation are discussed and numerical results are presented for Morse, Born-Meyer and Lennard-Jones potentials. Askar’s paper contains several misprints and errors and the corrected forms of his equations are presented in the appendix.

scholarly journals Spatial Analyticity of Solutions to Korteweg–de Vries Type Equations

2021 ◽  
Vol 26 (4) ◽  
pp. 75
Author(s):  
Keltoum Bouhali ◽  
Abdelkader Moumen ◽  
Khadiga W. Tajer ◽  
Khdija O. Taha ◽  
Yousif Altayeb
Keyword(s):  
Differential Equation ◽  
Nonlinear Waves ◽  
Vries Equation ◽  
Nonlinear Partial Differential Equation ◽  
Holomorphic Functions ◽  
Well Posedness ◽  
Initial Value ◽  
Third Order ◽  
De Vries ◽  
Type Spaces

The Korteweg–de Vries equation (KdV) is a mathematical model of waves on shallow water surfaces. It is given as third-order nonlinear partial differential equation and plays a very important role in the theory of nonlinear waves. It was obtained by Boussinesq in 1877, and a detailed analysis was performed by Korteweg and de Vries in 1895. In this article, by using multi-linear estimates in Bourgain type spaces, we prove the local well-posedness of the initial value problem associated with the Korteweg–de Vries equations. The solution is established online for analytic initial data w0 that can be extended as holomorphic functions in a strip around the x-axis. A procedure for constructing a global solution is proposed, which improves upon earlier results.

scholarly journals Asymptotic solutions of soliton type of the Korteweg–de Vries equation with variable coefficients and singular perturbation

2019 ◽  
Vol 6 (2) ◽  
pp. 374-385 ◽  
Author(s):  
V. H. Samoilenko ◽  
◽  
Yu. I. Samoilenko ◽  
V. O. Limarchenko ◽  
V. S. Vovk ◽  
...  
Keyword(s):  
Singular Perturbation ◽  
Vries Equation ◽  
Variable Coefficients ◽  
Asymptotic Solutions ◽  
De Vries

scholarly journals Solutions to the modified Korteweg–de Vries equation

2014 ◽  
Vol 26 (07) ◽  
pp. 1430006 ◽  
Author(s):  
Da-Jun Zhang ◽  
Song-Lin Zhao ◽  
Ying-Ying Sun ◽  
Jing Zhou
Keyword(s):  
Differential Equation ◽  
Vries Equation ◽  
Complete Solution ◽  
Matrix Differential Equation ◽  
Rational Solutions ◽  
Entry Vector ◽  
Complex Operation ◽  
De Vries ◽  
The Matrix ◽  
Matrix Differential

This is a continuation of [Notes on solutions in Wronskian form to soliton equations: Korteweg–de Vries-type, arXiv:nlin.SI/0603008]. In the present paper, we review solutions to the modified Korteweg–de Vries equation in terms of Wronskians. The Wronskian entry vector needs to satisfy a matrix differential equation set which contains complex operation. This fact makes the analysis of the modified Korteweg–de Vries to be different from the case of the Korteweg–de Vries equation. To derive complete solution expressions for the matrix differential equation set, we introduce an auxiliary matrix to deal with the complex operation. As a result, the obtained solutions to the modified Korteweg–de Vries equation are categorized into two types: solitons and breathers, together with their limit cases. Besides, we give rational solutions to the modified Korteweg–de Vries equation in Wronskian form. This is derived with the help of a Galilean transformed version of the modified Korteweg–de Vries equation. Finally, typical dynamics of the obtained solutions are analyzed and illustrated. We also list out the obtained solutions and their corresponding basic Wronskian vectors in the conclusion part.

Sign in / Sign up

Export Citation Format

Share Document