Using Vector-Product Loop Algebra to Generate Integrable Systems

A new three-dimensional Lie algebra and its loop algebra are proposed by us, whose commutator is a vector product. Based on this, a positive flow and a negative flow are obtained by introducing a new kind of spectral problem expressed by the vector product, which reduces to a generalized KdV equatio...

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Bibliographic Details
Published in:Axioms 2023-09, Vol.12 (9), p.840
Main Authors: Zhang, Jian, Feng, Binlu, Zhang, Yufeng, Ju, Long
Format: Article
Language:English
Subjects:
Algebra
Commutators
Equations
Hamiltonian functions
Hierarchies
integrable hierarchy
Lie algebra
Lie algebras
Lie groups
Linear algebra
Mathematical research
Schrodinger equation
zero-curvature equation
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Summary:A new three-dimensional Lie algebra and its loop algebra are proposed by us, whose commutator is a vector product. Based on this, a positive flow and a negative flow are obtained by introducing a new kind of spectral problem expressed by the vector product, which reduces to a generalized KdV equation, a generalized Schrödinger equation, a sine-Gordon equation, and a sinh-Gordon equation. Next, the well-known Tu scheme is generalized for generating isospectral integrable hierarchies and non-isospectral integrable hierarchies. It is important that we make use of the variational method to create a new vector-product trace identity for which the Hamiltonian structure of the isospectral integrable hierarchy presented in the paper is worded out. Finally, we further enlarge the three-dimensional loop algebra into a six-dimensional loop algebra so that a new isospectral integrable hierarchy which is a type of extended integrable model is produced whose bi-Hamiltonian structure is also derived from the vector-product trace identity. This new approach presented in the paper possesses extensive applications in the aspect of generating integrable hierarchies of evolution equations.
ISSN:2075-1680
2075-1680
DOI:10.3390/axioms12090840