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.

Reductions and Solutions of Two Types of Coupled Nonlinear Evolution Equations in Optical Fibers and Fluid Dynamics

2011 ◽  
Vol 66 (8-9) ◽  
pp. 552-558
Author(s):  
Li-Cai Liu ◽  
Bo Tian ◽  
Bo Qin ◽  
Xing Lü ◽  
Zhi-Qiang Lin ◽  
...  
Keyword(s):  
Fluid Dynamics ◽  
Raman Scattering ◽  
Optical Fibers ◽  
Evolution Equations ◽  
Nonlinear Evolution ◽  
Dark Soliton ◽  
Nonlinear Evolution Equations ◽  
The One ◽  
Gain Loss ◽  
Burgers Type Equations

Abstract Under investigation in this paper are the coupled nonlinear Schrödinger equations (CNLSEs) and coupled Burgers-type equations (CBEs), which are, respectively, a model for certain birefringent optical fibers Raman-scattering, Kerr and gain/loss effects, and a generalized model in fluid dynamics. Special attention should be paid to the existing claim that the solitons for the CNLSEs do not exist. Through certain dependent-variable transformations, the CNLSEs are reduced to a Manakov system and the CBEs are linearized. In that way, some new solutions of the CNLSEs and CBEs are obtained via symbolic computation. Especially the one-dark-soliton-like solutions for the CNLSEs have been found, against the existing claim.

scholarly journals Comment on “Conservation Laws of Two (2 + 1)-Dimensional Nonlinear Evolution Equations with Higher-Order Mixed Derivatives”

2014 ◽  
Vol 2014 ◽  
pp. 1-4
Author(s):  
Long Wei ◽  
Yang Wang
Keyword(s):  
Conservation Laws ◽  
Nonlinear Equations ◽  
Evolution Equations ◽  
Nonlinear Evolution ◽  
Higher Order ◽  
Nonlinear Evolution Equations ◽  
Symmetric Form ◽  
Mixed Derivatives ◽  
The One

In a recent paper (Zhang (2013)), the author claims that he has proposed two rules to modify Ibragimov’s theorem on conservation laws to “ensure the theorem can be applied to nonlinear evolution equations with any mixed derivatives.” In this letter, we analysis the paper. Indeed, the so-called “modification rules” are needless and the theorem of Ibragimov can be applied to construct conservation laws directly for nonlinear equations with any mixed derivatives as long as the formal Lagrangian is rewritten in symmetric form. Moreover, the conservation laws obtained by the so-called “modification rules” in the paper under discussion are equivalent to the one obtained by Ibragimov’s theorem.

Application of Extended Transformed Rational Function Method to Some (3+1) Dimensional Nonlinear Evolution Equations

2018 ◽  
pp. 433-437
Keyword(s):  
Rational Function ◽  
Function Method ◽  
Evolution Equations ◽  
Nonlinear Evolution ◽  
Kdv Equation ◽  
Mapping Method ◽  
Nonlinear Evolution Equations ◽  
Petviashvili Equation ◽  
Complexiton Solutions ◽  
Transformed Rational Function Method

The transformed rational function method can be considered as unification of the tanh type methods, the homogeneous balance method, the mapping method, the exp-function method and the F-expansion type methods. In this paper, we present complexiton solutions of (3+1) dimensional Korteweg-de Vries (KdV) equation and a new (3+1) dimensional generalized Kadomtsev-Petviashvili equation by using extended transformed rational function method which provides very useful and effective way to obtain complexiton solutions of nonlinear evolution equations.

On Nonlinear Equations Amenable to the Inverse Scattering Method

1975 ◽  
Vol 53 (1) ◽  
pp. 58-61 ◽  
Author(s):  
J. G. Kingston ◽  
C. Rogers
Keyword(s):  
Inverse Scattering ◽  
Initial Value Problem ◽  
Evolution Equations ◽  
Nonlinear Evolution ◽  
Inverse Scattering Method ◽  
Nonlinear Evolution Equations ◽  
Scattering Method ◽  
Initial Value ◽  
Physical Importance ◽  
Extensive Class

The inverse scattering method can be used to solve the initial value problem for various nonlinear evolution equations of physical importance. Here an extensive class of equations for which the technique is available is delimited.

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