scholarly journals Nonlinear Stability of Periodic Traveling Wave Solutions of the Generalized Korteweg–de Vries Equation

2009 ◽  
Vol 41 (5) ◽  
pp. 1921-1947 ◽  
Author(s):  
Mathew A. Johnson
Keyword(s):  
Nonlinear Stability ◽  
Traveling Wave ◽  
Vries Equation ◽  
Traveling Wave Solutions ◽  
Wave Solutions ◽  
De Vries ◽  
Periodic Traveling Wave

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.

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