scholarly journals Traveling wave solutions of degenerate coupled Korteweg-de Vries equation

2014 ◽  
Vol 55 (9) ◽  
pp. 091501 ◽  
Author(s):  
Metin Gürses ◽  
Aslı Pekcan
Keyword(s):  
Traveling Wave ◽  
Vries Equation ◽  
Traveling Wave Solutions ◽  
Wave Solutions ◽  
De Vries

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.

On exact traveling-wave solutions for local fractional Korteweg-de Vries equation

2016 ◽  
Vol 26 (8) ◽  
pp. 084312 ◽  
Author(s):  
Xiao-Jun Yang ◽  
J. A. Tenreiro Machado ◽  
Dumitru Baleanu ◽  
Carlo Cattani
Keyword(s):  
Traveling Wave ◽  
Vries Equation ◽  
Traveling Wave Solutions ◽  
Exact Traveling Wave Solutions ◽  
Wave Solutions ◽  
De Vries

scholarly journals Fractal complex transform technology for fractal Kkorteweg-de Vries equation within a local fractional derivative

2016 ◽  
Vol 20 (suppl. 3) ◽  
pp. 841-845
Author(s):  
Jinze Xu ◽  
Zeng-Shun Chen ◽  
Jian-Hong Wang ◽  
Ping Cui ◽  
Yunru Bai
Keyword(s):  
Fractional Derivative ◽  
Traveling Wave ◽  
Vries Equation ◽  
Special Functions ◽  
Traveling Wave Solutions ◽  
Cantor Sets ◽  
Local Fractional Derivative ◽  
Linear Partial Differential Equations ◽  
Wave Solutions ◽  
De Vries

In this paper, we present the fractal complex transform via a local fractional derivative. The traveling wave solutions for the fractal Korteweg-de Vries equations within local fractional derivative are obtained based on the special functions defined on Cantor sets. The technology is a powerful tool for solving the local fractional non-linear partial differential equations.

EXACT TRAVELING-WAVE SOLUTIONS FOR ONE-DIMENSIONAL MODIFIED KORTEWEG–DE VRIES EQUATION DEFINED ON CANTOR SETS

Fractals ◽  
2019 ◽  
Vol 27 (01) ◽  
pp. 1940010 ◽  
Author(s):  
FENG GAO ◽  
XIAO-JUN YANG ◽  
YANG JU
Keyword(s):  
Traveling Wave ◽  
Vries Equation ◽  
Traveling Wave Solutions ◽  
Mathematical Physics ◽  
Cantor Sets ◽  
Wave Transformation ◽  
One Dimensional ◽  
Wave Solutions ◽  
De Vries ◽  
The One

The one-dimensional modified Korteweg–de Vries equation defined on a Cantor set involving the local fractional derivative is investigated in this paper. With the aid of the fractal traveling-wave transformation technology, the nondifferentiable traveling-wave solutions for the problem are discussed in detail. The obtained results are accurate and efficient for describing the fractal water wave in mathematical physics.

scholarly journals Traveling wave solutions for the extended modified KORTEWEG-DE VRIES equation

2021 ◽  
Vol 82 (4) ◽  
pp. 197-203
Author(s):  
G. N. Shaikhova ◽  
◽  
B. K. Rakhimzhanov ◽  
Keyword(s):  
Traveling Wave ◽  
Vries Equation ◽  
Nonlinear Partial Differential Equations ◽  
Integrable Model ◽  
Soliton Solutions ◽  
Traveling Wave Solutions ◽  
Mathematical Tool ◽  
Wave Solutions ◽  
De Vries ◽  
Fifth Order

In this paper, we study an extended modified Korteweg-de Vries equation, which contains the relevant higher-order nonlinear terms and fifth-order dispersion. This equation is the extension of the modified Korteweg-de Vries equation and described by the Ablowitz-Kaup-Newell-Segur hierarchy. The standard Korteweg-de Vries equation is the pioneer integrable model in solitary waves theory, which gives rise to multiple soliton solutions. The Korteweg-de Vries equation arises naturally from shallow water, plasma physics, and other fields of science. To obtain exact solutions the sine-cosine method is applied. It is shown that the sine-cosine method provides a powerful mathematical tool for solving a great many nonlinear partial differential equations in mathematical physics. Traveling wave solutions are determined for extended modified Korteweg-de Vries equation. The study shows that the sine–cosine method is quite efficient and practically well suited for use in calculating traveling wave solutions for extended modified Korteweg-de Vries equation.

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