Coupling of completely integrable systems: the perturbation bundle
Benno Fuchssteiner
- pp 125-138
TLDR
In this article, a canonical Lie algebra in the direct sum of vector fields and 1-1-tensors, the perturbation bundle, is introduced and extended to a full tensor structure and it is related to the Lie algebra obtained by coupling linear systems to nonlinear ones.Abstract:
We introduce a canonical Lie algebra in the direct sum of vector fields and 1-1-tensors, the perturbation bundle This Lie algebra is extended to a full tensor structure and it is related to the Lie algebra obtained by coupling linear systems to nonlinear ones Using Lie algebra isomorphisms from the original structure to the abstract perturbation bundle, new completely integrable systems are obtained The formalism of Lax pairs is found to be a special case of the new structureread more
Citations
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Integrable theory of the perturbation equations
Wen-Xiu Ma,Benno Fuchssteiner +1 more
TL;DR: In this paper, an integrable theory for perturbation equations engendered from small disturbances of solutions is developed, which includes various integrability properties of the perturbations, such as hereditary recursion operators, master symmetries, linear representations (Lax and zero curvature representations) and Hamiltonian structures, and provides us with a method of generating hereditary operators, Hamiltonian operators and symplectic operators starting from the known ones.
Journal ArticleDOI
The quadratic-form identity for constructing the Hamiltonian structure of integrable systems
TL;DR: In this article, a quadratic-form identity for linear isospectral integrable systems is presented, whose special case is just the trace identity; that is, when taking the loop algebra, the quadratically-form identities presented in this paper is completely consistent with the trace identities.
Journal ArticleDOI
A direct method for integrable couplings of TD hierarchy
Yufeng Zhang,Hongqing Zhang +1 more
TL;DR: In this paper, a direct method for establishing integrable couplings of TD hierarchy is proposed, which is obtained by constructing a suitable transformation of Lax pairs and a new Lie algebra.
Journal ArticleDOI
Integrable couplings of soliton equations by perturbations I: A general theory and application to the KDV hierarchy
TL;DR: In this paper, a theory for constructing integrable couplings of soliton equations is developed by using various perturbations around solutions of perturbed soliton equation being analytic with respect to a small perturbation parameter.
Journal ArticleDOI
Integrable couplings of vector AKNS soliton equations
TL;DR: By enlarging the associated matrix spectral problems, two specific classes of multicomponent integrable couplings of the physically important vector AKNS soliton equations are constructed, which can be linked to each other by a Backlund transformation.
References
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TL;DR: In this paper, the authors studied the permanence properties of hereditary operators in Lie algebras and showed that they can be interpreted as special Lie algebra deformations with a linear interpolation property.
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TL;DR: In this article, a method of constructing Lax representations for isospectral and nonispectral hierarchies of evolution equations is proposed, and it is shown that under some simple but essential conditions, the corresponding Lax operators may constitute infinite-dimensional Lie algebras with respect to the binary operation.