Non-simple blow-up solutions for the Neumann two-dimensional sinh-Gordon equation

2008 ◽  
Vol 34 (3) ◽  
pp. 341-375 ◽  
Author(s):  
Pierpaolo Esposito ◽  
Juncheng Wei
Keyword(s):  
Gordon Equation ◽  
Blow Up ◽  
Two Dimensional

scholarly journals Duality of positive and negative integrable hierarchies via relativistically invariant fields

2021 ◽  
Vol 2021 (7) ◽  
Author(s):  
S. Y. Lou ◽  
X. B. Hu ◽  
Q. P. Liu
Keyword(s):  
Integrable Hierarchies ◽  
Gordon Equation ◽  
Formal Series ◽  
Two Dimensional ◽  
Relativistic Invariance ◽  
Sine Gordon Equation ◽  
Recursion Operators ◽  
Symmetry Approach ◽  
Duality Relations ◽  
Duality Problem

Abstract It is shown that the relativistic invariance plays a key role in the study of integrable systems. Using the relativistically invariant sine-Gordon equation, the Tzitzeica equation, the Toda fields and the second heavenly equation as dual relations, some continuous and discrete integrable positive hierarchies such as the potential modified Korteweg-de Vries hierarchy, the potential Fordy-Gibbons hierarchies, the potential dispersionless Kadomtsev-Petviashvili-like (dKPL) hierarchy, the differential-difference dKPL hierarchy and the second heavenly hierarchies are converted to the integrable negative hierarchies including the sG hierarchy and the Tzitzeica hierarchy, the two-dimensional dispersionless Toda hierarchy, the two-dimensional Toda hierarchies and negative heavenly hierarchy. In (1+1)-dimensional cases the positive/negative hierarchy dualities are guaranteed by the dualities between the recursion operators and their inverses. In (2+1)-dimensional cases, the positive/negative hierarchy dualities are explicitly shown by using the formal series symmetry approach, the mastersymmetry method and the relativistic invariance of the duality relations. For the 4-dimensional heavenly system, the duality problem is studied firstly by formal series symmetry approach. Two elegant commuting recursion operators of the heavenly equation appear naturally from the formal series symmetry approach so that the duality problem can also be studied by means of the recursion operators.

An improved blow-up criterion for smooth solutions of the two-dimensional MHD equations

2016 ◽  
Vol 40 (1) ◽  
pp. 279-285 ◽  
Author(s):  
Sadek Gala ◽  
Alessandra Maria Ragusa ◽  
Zhuan Ye
Keyword(s):  
Blow Up ◽  
Mhd Equations ◽  
Two Dimensional ◽  
Smooth Solutions ◽  
Blow Up Criterion
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