# zbMATH — the first resource for mathematics

Two types of new Lie algebras and corresponding hierarchies of evolution equations. (English) Zbl 1011.17018
Summary: An extension of the Lie algebra $$A_{n-1}$$ is proposed. As special cases, two new loop algebras are constructed, respectively. It follows that two types of new integrable Hamiltonian hierarchies are engendered. As their reduction cases, generalized nonlinear Schrödinger equations, coupled Fisher equations, and the standard heat-conduction equation are obtained, respectively. The method proposed can be used generally.

##### MSC:
 17B67 Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras 37K30 Relations of infinite-dimensional Hamiltonian and Lagrangian dynamical systems with infinite-dimensional Lie algebras and other algebraic structures 35Q55 NLS equations (nonlinear Schrödinger equations) 35K05 Heat equation 17B80 Applications of Lie algebras and superalgebras to integrable systems 37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)
Full Text:
##### References:
 [1] Tu, G., J. math. phys., 30, 2, 330, (1989) [2] W. Ma, Ph.D. Dissertation, Academia Sinica, Beijing, China, 1990 [3] Hu, X., J. phys. A: math. gen., 30, 619, (1997) [4] Fan, E., J. math. phys., 41, 11, 7769, (2000) [5] Gu, C., Soliton theory and its application, (1990), Zhejiang Publishing House of Science and Technology, (in Chinese)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.