Conservation laws of some multi-component integrable systems

2021 ◽  
pp. 2150405
Author(s):  
Linlin Gui ◽  
Yufeng Zhang
Keyword(s):  
Conservation Laws ◽  
Integrable Systems ◽  
The Self ◽  
Component Systems

Some multi-component integrable systems are introduced and constructed. These multi-component systems are selected as examples to help us study the self-adjointness of [Formula: see text] Frobenius equation by means of some new definitions and theorems that we introduced. It follows that the conservation laws of these multi-component systems are constructed based on a few symmetries.

scholarly journals New cosmological solutions in hybrid metric-Palatini gravity from dynamical symmetries

2021 ◽  
pp. 2150100
Author(s):  
Andronikos Paliathanasis
Keyword(s):  
Conservation Laws ◽  
Integrable Systems ◽  
Functional Form ◽  
Analytic Solutions ◽  
Momentum Conservation ◽  
Field Equations ◽  
Dynamical Symmetries ◽  
Cosmological Solutions ◽  
Liouville Integrable

We investigate the existence of Liouville integrable cosmological models in hybrid metric-Palatini theory. Specifically, we use the symmetry conditions for the existence of quadratic in the momentum conservation laws for the field equations as constraint conditions for the determination of the unknown functional form of the theory. The exact and analytic solutions of the integrable systems found in this study are presented in terms of quadratics and Laurent expansions.

(1+1)-dimensional m-cKdV, g-cKdV integrable systems, and (2+1)-dimensional m-cKdV hierarchy

2008 ◽  
Vol 86 (12) ◽  
pp. 1367-1380 ◽  
Author(s):  
Y Zhang ◽  
H Tam
Keyword(s):  
Integrable Systems ◽  
Linear Functional ◽  
Hamiltonian Structure ◽  
Evolution Equations ◽  
Integrable Model ◽  
Lax Pair ◽  
The Self ◽  
Hamiltonian Structures ◽  
Yang Mills ◽  
Higher Dimensional

A few isospectral problems are introduced by referring to that of the cKdV equation hierarchy, for which two types of integrable systems called the (1 + 1)-dimensional m-cKdV hierarchy and the g-cKdV hierarchy are generated, respectively, whose Hamiltonian structures are also discussed by employing a linear functional and the quadratic-form identity. The corresponding expanding integrable models of the (1 + 1)-dimensional m-cKdV hierarchy and g-cKdV hierarchy are obtained. The Hamiltonian structure of the latter one is given by the variational identity, proposed by Ma Wen-Xiu as well. Finally, we use a Lax pair from the self-dual Yang–Mills equations to deduce a higher dimensional m-cKdV hierarchy of evolution equations and its Hamiltonian structure. Furthermore, its expanding integrable model is produced by the use of a enlarged Lie algebra.PACS Nos.: 02.30, 03.40.K

scholarly journals Self-Adjointness and Conservation Laws of Frobenius Type Equations

Symmetry ◽  
2020 ◽  
Vol 12 (12) ◽  
pp. 1987
Author(s):  
Haifeng Wang ◽  
Yufeng Zhang
Keyword(s):  
Conservation Laws ◽  
Burgers Equation ◽  
Kdv Equation ◽  
The Self ◽  
Kp Equation ◽  
Commutative Subalgebra ◽  
Kompaneets Equation ◽  
Dym Equation ◽  
Harry Dym Equation

The Frobenius KDV equation and the Frobenius KP equation are introduced, and the Frobenius Kompaneets equation, Frobenius Burgers equation and Frobenius Harry Dym equation are constructed by taking values in a commutative subalgebra Z2ε in the paper. The five equations are selected as examples to help us study the self-adjointness of Frobenius type equations, and we show that the first two equations are quasi self-adjoint and the last three equations are nonlinear self-adjointness. It follows that we give the symmetries of the Frobenius KDV and the Frobenius KP equation in order to construct the corresponding conservation laws.

scholarly journals Multi‐dimensional conservation laws and integrable systems

2019 ◽  
Vol 143 (4) ◽  
pp. 339-355
Author(s):  
Zakhar V. Makridin ◽  
Maxim V. Pavlov
Keyword(s):  
Conservation Laws ◽  
Integrable Systems

Conservation Laws of the Self-adjoint K(m,n) Equation with Generalized Evolution Term

2011 ◽  
Author(s):  
M. S. Bruzón ◽  
M. L. Gandarias ◽  
Theodore E. Simos ◽  
George Psihoyios ◽  
Ch. Tsitouras ◽  
...  
Keyword(s):  
Conservation Laws ◽  
The Self

scholarly journals A super Degasperis–Procesi equation and related integrable systems

2021 ◽  
Vol 477 (2245) ◽  
pp. 20200780
Author(s):  
Binfang Gao ◽  
Kai Tian ◽  
Qing Ping Liu
Keyword(s):  
Conservation Laws ◽  
Spectral Problem ◽  
Integrable Systems ◽  
Hamiltonian Structure ◽  
Reciprocal Transformation ◽  
Matrix Spectral Problem

Based on a 4 × 4 matrix spectral problem, a super Degasperis–Procesi (DP) equation is proposed. We show that under a reciprocal transformation, the super DP equation is related to the first negative flow of a super Kaup–Kupershmidt (KK) hierarchy, which turns out to be a particular reduction of a super Boussinesq hierarchy. The bi-Hamiltonian structure of the super Boussinesq hierarchy is established and subsequently produces a Hamiltonian structure, as well as a conjectured symplectic formulation of the super KK hierarchy via suitable reductions. With the help of the reciprocal transformation, the bi-Hamiltonian representation of the super DP equation is constructed from that of the super KK hierarchy. We also calculate a positive flow of the super DP hierarchy and explain its relations with the super KK equation. Infinitely many conservation laws are derived for the super DP equation, as well as its positive flow.

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