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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.)
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References:
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