A new approach for generating the TX hierarchy as well as its integrable couplings. (English) Zbl 1472.37073

Summary: G. Tu and B. Xu [Chin. Ann. Math., Ser. B 17, No. 4, 497–506 (1996; Zbl 0909.70020)] once introduced an isospectral problem by a loop algebra with degree being \(\lambda\), for which an integrable hierarchy of evolution equations (called the TX hierarchy) was derived under the frame of zero curvature equations. In the paper, we present a loop algebra whose degrees are \(2 \lambda\) and \(2 \lambda + 1\) to simply represent the above isospectral matrix and easily derive the TX hierarchy. Specially, through enlarging the loop algebra with 3 dimensions to 6 dimensions, we generate a new integrable coupling of the TX hierarchy and its corresponding Hamiltonian structure.


37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)
37K30 Relations of infinite-dimensional Hamiltonian and Lagrangian dynamical systems with infinite-dimensional Lie algebras and other algebraic structures
37K06 General theory of infinite-dimensional Hamiltonian and Lagrangian systems, Hamiltonian and Lagrangian structures, symmetries, conservation laws


Zbl 0909.70020
Full Text: DOI


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