New Dark Soliton Solutions of the KdV Equation

1993 ◽  
Vol 20 (4) ◽  
pp. 493-493
Author(s):  
Zong-Yun Chen ◽  
Nian-Ning Huang
Keyword(s):  
Kdv Equation ◽  
Soliton Solutions ◽  
Dark Soliton ◽  
The Kdv Equation

scholarly journals Fractional Soliton Dynamics and Spectral Transform of Time-Fractional Nonlinear Systems: A Concrete Example

Complexity ◽  
2019 ◽  
Vol 2019 ◽  
pp. 1-9 ◽  
Author(s):  
Sheng Zhang ◽  
Yuanyuan Wei ◽  
Bo Xu
Keyword(s):  
Exact Solution ◽  
Fractional Derivative ◽  
Spectral Problem ◽  
Local Time ◽  
Kdv Equation ◽  
Soliton Solutions ◽  
Soliton Dynamics ◽  
Spectral Transform ◽  
The Kdv Equation ◽  
Time Fractional Derivative

In this paper, the spectral transform with the reputation of nonlinear Fourier transform is extended for the first time to a local time-fractional Korteweg-de vries (tfKdV) equation. More specifically, a linear spectral problem associated with the KdV equation of integer order is first equipped with local time-fractional derivative. Based on the spectral problem with the equipped local time-fractional derivative, the local tfKdV equation with Lax integrability is then derived and solved by extending the spectral transform. As a result, a formula of exact solution with Mittag-Leffler functions is obtained. Finally, in the case of reflectionless potential the obtained exact solution is reduced to fractional n-soliton solution. In order to gain more insights into the fractional n-soliton dynamics, the dynamical evolutions of the reduced fractional one-, two-, and three-soliton solutions are simulated. It is shown that the velocities of the reduced fractional one-, two-, and three-soliton solutions change with the fractional order.

scholarly journals The static elliptic N-soliton solutions of the KdV equation

2019 ◽  
Vol 3 (4) ◽  
pp. 045004
Author(s):  
Masahito Hayashi ◽  
Kazuyasu Shigemoto ◽  
Takuya Tsukioka
Keyword(s):  
Kdv Equation ◽  
Soliton Solutions ◽  
The Kdv Equation

scholarly journals A Generalized KdV Equation of Neglecting the Highest-Order Infinitesimal Term and Its Exact Traveling Wave Solutions

2013 ◽  
Vol 2013 ◽  
pp. 1-15 ◽  
Author(s):  
Xianbin Wu ◽  
Weiguo Rui ◽  
Xiaochun Hong
Keyword(s):  
Traveling Wave ◽  
Kdv Equation ◽  
Dynamic Properties ◽  
Soliton Solutions ◽  
Traveling Wave Solutions ◽  
Wave Model ◽  
Dark Soliton ◽  
Exact Traveling Wave Solutions ◽  
Generalized Kdv Equation ◽  
Wave Solutions

We study a generalized KdV equation of neglecting the highest order infinitesimal term, which is an important water wave model. Some exact traveling wave solutions such as singular solitary wave solutions, semiloop soliton solutions, dark soliton solutions, dark peakon solutions, dark loop-soliton solutions, broken loop-soliton solutions, broken wave solutions of U-form and C-form, periodic wave solutions of singular type, and broken wave solution of semiparabola form are obtained. By using mathematical softwareMaple, we show their profiles and discuss their dynamic properties. Investigating these properties, we find that the waveforms of some traveling wave solutions vary with changes of certain parameters.

Dark Solitons for a Generalized Korteweg-de Vries Equation with Time-Dependent Coefficients

2011 ◽  
Vol 66 (3-4) ◽  
pp. 199-204
Author(s):  
Houria Triki ◽  
Abdul-Majid Wazwaz
Keyword(s):  
Pulse Propagation ◽  
Kdv Equation ◽  
Critical Rayleigh Number ◽  
Soliton Solutions ◽  
Dark Soliton ◽  
Time Dependent ◽  
Ansatz Method ◽  
Wave Dynamics ◽  
De Vries ◽  
Dependent Model

We consider the evolution of long shallow waves in a convecting fluid when the critical Rayleigh number slightly exceeds its critical value within the framework of a perturbed Korteweg-de Vries (KdV) equation. In order to study the wave dynamics of nonlinear pulse propagation in an inhomogeneous KdV media, a generalized form of the considered model with time-dependent coefficients is presented. By means of the solitary wave ansatz method, exact dark soliton solutions are derived under certain parametric conditions. The results show that the soliton parameters (amplitude, inverse width, and velocity) are influenced by the time variation of the dependent model coefficients. The existence of such a soliton solution is the result of the exact balance among nonlinearity, third-order and fourth-order nonlinear dispersions, diffusion, dissipation, and reaction.

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