A group-theoretical derivation of the conservation laws for nonlinear evolution equations

1980 ◽  
Vol 76 (3-4) ◽  
pp. 209-210
Author(s):  
B.L. Markovski ◽  
V.A. Rizov
Keyword(s):  
Conservation Laws ◽  
Evolution Equations ◽  
Nonlinear Evolution ◽  
Nonlinear Evolution Equations ◽  
Theoretical Derivation

scholarly journals Conservation Laws of Nonlinear-Evolution Equations

1974 ◽  
Vol 52 (3) ◽  
pp. 886-889 ◽  
Author(s):  
K. Konno ◽  
H. Sanuki ◽  
Y. H. Ichikawa
Keyword(s):  
Conservation Laws ◽  
Evolution Equations ◽  
Nonlinear Evolution ◽  
Nonlinear Evolution Equations

An extension of the coupled derivative nonlinear Schrödinger hierarchy

2018 ◽  
Vol 32 (02) ◽  
pp. 1850016
Author(s):  
Siqi Xu ◽  
Xianguo Geng ◽  
Bo Xue
Keyword(s):  
Conservation Laws ◽  
Spectral Problem ◽  
Compatibility Condition ◽  
Evolution Equations ◽  
Nonlinear Evolution ◽  
Nonlinear Evolution Equations ◽  
Trace Identity ◽  
Hamiltonian Structures ◽  
Nonlinear Schrödinger ◽  
Matrix Spectral Problem

In this paper, a 3 × 3 matrix spectral problem with six potentials is considered. With the help of the compatibility condition, a hierarchy of new nonlinear evolution equations which can be reduced to the coupled derivative nonlinear Schrödinger (CDNLS) equations is obtained. By use of the trace identity, it is proved that all the members in this new hierarchy have generalized bi-Hamiltonian structures. Moreover, infinitely many conservation laws of this hierarchy are constructed.

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.

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