Padé approximants and soliton solutions of the KdV equation

1980 ◽  
Vol 27 (4) ◽  
pp. 107-110 ◽  
Author(s):  
G. Turchetti
Keyword(s):  
Kdv Equation ◽  
Soliton Solutions ◽  
Padé Approximants ◽  
Pade Approximants ◽  
The Kdv Equation

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

Integrable Models

2018 ◽  
Author(s):  
S. G. Rajeev
Keyword(s):  
Fluid Mechanics ◽  
Soliton Solution ◽  
Water Waves ◽  
Heisenberg Model ◽  
Kdv Equation ◽  
Short Review ◽  
Lax Pair ◽  
Shallow Water Waves ◽  
Initial Value ◽  
The Kdv Equation

Some exceptional situations in fluid mechanics can be modeled by equations that are analytically solvable. The most famous example is the Korteweg–de Vries (KdV) equation for shallow water waves in a channel. The exact soliton solution of this equation is derived. The Lax pair formalism for solving the general initial value problem is outlined. Two hamiltonian formalisms for the KdV equation (Fadeev–Zakharov and Magri) are explained. Then a short review of the geometry of curves (Frenet–Serret equations) is given. They are used to derive a remarkably simple equation for the propagation of a kink along a vortex filament. This equation of Hasimoto has surprising connections to the nonlinear Schrödinger equation and to the Heisenberg model of ferromagnetism. An exact soliton solution is found.

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