On the inverse scattering transform, the cylindrical Korteweg-De Vries equation and similarity solutions

1979 ◽  
Vol 72 (3) ◽  
pp. 197-199 ◽  
Author(s):  
R.S. Johnson
Keyword(s):  
Inverse Scattering ◽  
Vries Equation ◽  
Inverse Scattering Transform ◽  
Similarity Solutions ◽  
Scattering Transform ◽  
De Vries

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.

scholarly journals Inverse scattering transform for a supersymmetric Korteweg-de Vries equation

2019 ◽  
Vol 23 (Suppl. 3) ◽  
pp. 677-684
Author(s):  
Sheng Zhang ◽  
Caihong You
Keyword(s):  
Exact Solutions ◽  
Inverse Scattering ◽  
Evolution Equations ◽  
Vries Equation ◽  
Soliton Solutions ◽  
Grassmann Algebra ◽  
Inverse Scattering Transform ◽  
Linear Evolution ◽  
Scattering Transform ◽  
De Vries

In this paper, the inverse scattering transform is extended to a super Korteweg-de Vries equation with an arbitrary variable coefficient by using Kulish and Zeitlin?s approach. As a result, exact solutions of the super Korteweg-de Vries equation are obtained. In the case of reflectionless potentials, the obtained exact solutions are reduced to soliton solutions. More importantly, based on the obtained results, an approach to extending the scattering transform is proposed for the supersymmetric Korteweg-de Vries equation in the 1-D Grassmann algebra. It is shown the the approach can be applied to some other supersymmetric non-linear evolution equations in fluids.

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