On a class of coupled Hamiltonian operators and their integrable hierarchies with two potentials

2018 ◽  
Vol 41 (10) ◽  
pp. 3779-3789 ◽  
Author(s):  
Xiang Gu ◽  
Wen-Xiu Ma
Keyword(s):  
Integrable Hierarchies ◽  
Hamiltonian Operators

scholarly journals On the compatible weakly nonlocal Poisson brackets of hydrodynamic type

2002 ◽  
Vol 32 (10) ◽  
pp. 587-614 ◽  
Author(s):  
Andrei Ya. Maltsev
Keyword(s):  
Integrable Hierarchies ◽  
Riemannian Metrics ◽  
Riemannian Metric ◽  
Hydrodynamic Type ◽  
Poisson Brackets ◽  
Symplectic Structures ◽  
Symplectic Forms ◽  
Hamiltonian Functions ◽  
Hamiltonian Operators ◽  
Nonlocal Part

We consider the pairs of general weakly nonlocal Poisson brackets of hydrodynamic type (Ferapontov brackets) and the corresponding integrable hierarchies. We show that, under the requirement of the nondegeneracy of the corresponding “first” pseudo-Riemannian metricg(0) νμand also some nondegeneracy requirement for the nonlocal part, it is possible to introduce a “canonical” set of “integrable hierarchies” based on the Casimirs, momentum functional and some “canonical Hamiltonian functions.” We prove also that all the “higher” “positive” Hamiltonian operators and the “negative” symplectic forms have the weakly nonlocal form in this case. The same result is also true for “negative” Hamiltonian operators and “positive” symplectic structures in the case when both pseudo-Riemannian metricsg(0) νμandg(1) νμare nondegenerate.

scholarly journals Positive and Negative Integrable Hierarchies: Bi-Hamiltonian Structure and Darboux–Bäcklund Transformation

2020 ◽  
Vol 2020 ◽  
pp. 1-15
Author(s):  
Ning Zhang ◽  
Xi-Xiang Xu
Keyword(s):  
Hamiltonian Structure ◽  
Integrable Hierarchies ◽  
Bäcklund Transformation ◽  
Liouville Integrability ◽  
Backlund Transformation ◽  
Hamiltonian Structures ◽  
Negative Power ◽  
Zero Curvature ◽  
Matrix Spectral Problem ◽  
Hamiltonian Operators

Two integrable hierarchies are derived from a novel discrete matrix spectral problem by discrete zero curvature equations. They correspond, respectively, to positive power and negative power expansions of Lax operators with respect to the spectral parameter. The bi-Hamiltonian structures of obtained hierarchies are established by a pair of Hamiltonian operators through discrete trace identity. The Liouville integrability of the obtained hierarchies is proved. Through a gauge transformation of the Lax pair, a Darboux–Bäcklund transformation is constructed for the first nonlinear different-difference equation in the negative hierarchy. Ultimately, applying the obtained Darboux–Bäcklund transformation, two exact solutions are given by means of mathematical software.

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.

Loop algebra and visaoro symmetries of integrable hierarchies of KP type

2001 ◽  
Vol 78 (3-4) ◽  
pp. 233-253 ◽  
Author(s):  
H. Aratyn ◽  
J.F. Gomes ◽  
E. Nissimov ◽  
S. Pacheva
Keyword(s):  
Integrable Hierarchies ◽  
Loop Algebra

scholarly journals Intertwining operator and integrable hierarchies from topological strings

2021 ◽  
Vol 2021 (5) ◽  
Author(s):  
Jean-Emile Bourgine
Keyword(s):  
Topological Strings ◽  
Vertex Operator ◽  
Integrable Hierarchies ◽  
Kp Hierarchy ◽  
Tau Function ◽  
Time Deformation ◽  
Toroidal Algebra ◽  
Crystal Model ◽  
Melting Crystal ◽  
Definition Of

Abstract In [1], Nakatsu and Takasaki have shown that the melting crystal model behind the topological strings vertex provides a tau-function of the KP hierarchy after an appropriate time deformation. We revisit their derivation with a focus on the underlying quantum W1+∞ symmetry. Specifically, we point out the role played by automorphisms and the connection with the intertwiner — or vertex operator — of the algebra. This algebraic perspective allows us to extend part of their derivation to the refined melting crystal model, lifting the algebra to the quantum toroidal algebra of $$ \mathfrak{gl} $$ gl (1) (also called Ding-Iohara-Miki algebra). In this way, we take a first step toward the definition of deformed hierarchies associated to A-model refined topological strings.

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