Remarks on Nonlinear Evolution Equations and the Inverse Scattering Transform

1981 ◽  
pp. 83-94
Author(s):  
Mark J. Ablowitz
Keyword(s):  
Inverse Scattering ◽  
Evolution Equations ◽  
Nonlinear Evolution ◽  
Inverse Scattering Transform ◽  
Nonlinear Evolution Equations ◽  
Scattering Transform

ANALYTIC METHODS FOR SOLITON SYSTEMS

1993 ◽  
Vol 03 (01) ◽  
pp. 3-17 ◽  
Author(s):  
M. LAKSHMANAN
Keyword(s):  
Evolution Equations ◽  
Nonlinear Evolution ◽  
Soliton Solutions ◽  
Direct Methods ◽  
Nonlinear Evolution Equations ◽  
Initial Value ◽  
Mathematical Concepts ◽  
Transform Method ◽  
Analytic Methods ◽  
Scattering Transform

The study of soliton systems continues to be a highly rewarding exercise in nonlinear dynamics, even though it has been almost thirty years since the introduction of the soliton concept by Zabusky & Kruskal. Increasingly sophisticated mathematical concepts are being identified with integrable soliton systems, while newer applications are being made frequently. In this pedagogical review, after introducing solitons and their (2+1)-dimensional generalizations, we give an elementary discussion on the various analytic methods available for investigation of the soliton possessing nonlinear evolution equations. These include the inverse scattering transform method and its generalization, namely the d-bar approach, for solving the Cauchy initial value problem, as well as direct methods for obtaining N-soliton solutions. We also indicate how the Painlevé singularity structure analysis is useful for the detection of soliton systems.

scholarly journals On a direct construction of inverse scattering problems for integrable nonlinear evolution equations in the two-spatial dimension

2004 ◽  
Vol 2004 (58) ◽  
pp. 3117-3128
Author(s):  
H. H. Chen ◽  
J. E. Lin
Keyword(s):  
Inverse Scattering ◽  
Evolution Equations ◽  
Nonlinear Evolution ◽  
Spatial Dimension ◽  
Nonlinear Evolution Equations ◽  
Scattering Problems ◽  
Direct Construction ◽  
Inverse Scattering Problems ◽  
Petviashvili Equation ◽  
The One

We present a method to construct inverse scattering problems for integrable nonlinear evolution equations in the two-spatial dimension. The temporal component is the adjoint of the linearized equation and the spatial component is a partial differential equation with respect to the spatial variables. Although this idea has been known for the one-spatial dimension for some time, it is the first time that this method is presented for the case of the higher-spatial dimension. We present this method in detail for the Veselov-Novikov equation and the Kadomtsev-Petviashvili equation.

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