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 On the Korteweg-de Vries equation: an associated equation

1984 ◽  
Vol 7 (2) ◽  
pp. 263-277 ◽  
Author(s):  
Eugene P. Schlereth ◽  
Ervin Y. Rodin
Keyword(s):  
Initial Value Problems ◽  
Vries Equation ◽  
Kdv Equation ◽  
Nonlinear Partial Differential Equation ◽  
Inverse Scattering Transform ◽  
Similarity Solutions ◽  
Airy Functions ◽  
Initial Value ◽  
Scattering Transform ◽  
De Vries

The purpose of this paper is to describe a relationship between the Korteweg-de Vries (KdV) equationut−6uux+uxxx=0and another nonlinear partial differential equation of the formzt+zxxx−3zxzxxz=H(t)z.The second equation will be called the Associated Equation (AE) and the connection between the two will be explained. By considering AE, explicit solutions to KdV will be obtained. These solutions include the solitary wave and the cnoidal wave solutions. In addition, similarity solutions in terms of Airy functions and Painlevé transcendents are found. The approach here is different from the Inverse Scattering Transform and the results are not in the form of solutions to specific initial value problems, but rather in terms of solutions containing arbitrary constants.

Abundant rogue wave solutions for the (2 + 1)-dimensional generalized Korteweg–de Vries equation

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Huanhuan Lu ◽  
Yufeng Zhang
Keyword(s):  
Vries Equation ◽  
Rogue Wave ◽  
Rogue Waves ◽  
Third Order ◽  
First Order ◽  
Wave Solutions ◽  
De Vries ◽  
Rogue Wave Solutions ◽  
Bilinear Formulation ◽  
Hirota Bilinear

AbstractIn this paper, we analyse two types of rogue wave solutions generated from two improved ansatzs, to the (2 + 1)-dimensional generalized Korteweg–de Vries equation. With symbolic computation, the first-order rogue waves, second-order rogue waves, third-order rogue waves are generated directly from the first ansatz. Based on the Hirota bilinear formulation, another type of one-rogue waves and two-rogue waves can be obtained from the second ansatz. In addition, the dynamic behaviours of obtained rogue wave solutions are illustrated graphically.

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.

scholarly journals On the Korteweg–de Vries equation: Frequencies and initial value problem

1997 ◽  
Vol 181 (1) ◽  
pp. 1-55 ◽  
Author(s):  
D. Bättig ◽  
T. Kappeler ◽  
B. Mityagin
Keyword(s):  
Initial Value Problem ◽  
Vries Equation ◽  
Initial Value ◽  
De Vries

scholarly journals New Cubic B-spline Approximations for Solving Non-linear Third-order Korteweg-de Vries Equation

2019 ◽  
Vol 12 (6) ◽  
pp. 1-9
Author(s):  
Muhammad Abbas ◽  
Muhammad Kashif Iqbal ◽  
Bushra Zafar ◽  
Shazalina Binti Mat Zin ◽  
◽  
...  
Keyword(s):  
Vries Equation ◽  
Third Order ◽  
B Spline ◽  
Non Linear ◽  
De Vries

PROPAGATION OF WEAKLY NONLINEAR WAVES IN A FLUID-FILLED THIN ELASTIC TUBE

1999 ◽  
Vol 23 (2) ◽  
pp. 253-265
Author(s):  
H. Demiray
Keyword(s):  
Nonlinear Waves ◽  
Vries Equation ◽  
Perturbation Technique ◽  
Fluid Pressure ◽  
Static Field ◽  
Elastic Tube ◽  
Inviscid Fluid ◽  
Weakly Nonlinear ◽  
De Vries ◽  
Weakly Nonlinear Waves

In this work, we study the propagation of weakly nonlinear waves in a prestressed thin elastic tube filled with an inviscid fluid. In the analysis, considering the physiological conditions under which the arteries function, the tube is assumed to be subjected to a uniform pressure P0 and a constant axial stretch ratio λz. In the course of blood flow in arteries, it is assumed that a finite time dependent radial displacement is superimposed on this static field but, due to axial tethering, the effect of axial displacement is neglected. The governing nonlinear equation for the radial motion of the tube under the effect of fluid pressure is obtained. Using the exact nonlinear equations of an incompressible inviscid fluid and the reductive perturbation technique, the propagation of weakly nonlinear waves in a fluid-filled thin elastic tube is investigated in the longwave approximation. The governing equation for this special case is obtained as the Korteweg-de-Vries equation. It is shown that, contrary to the result of previous works on the same subject, in the present work, even for Mooney-Rivlin material, it is possible to obtain the nonlinear Korteweg-de-Vries equation.

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.

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