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Integrable deformations of the mKdV and SG hierarchies and quasigraded Lie algebras. (English) Zbl 1134.37359
Summary: We construct a new family of quasigraded Lie algebras that admit the Kostant–Adler scheme. They coincide with special quasigraded deformations of twisted subalgebras of the loop algebras. Using them we obtain new hierarchies of integrable equations in partial derivatives. They coincide with the deformations of integrable hierarchies associated with the loop algebras. We consider the case \(\mathfrak g = gl(2)\) in detail and obtain integrable hierarchies that could be viewed as deformations of mKdV, sine-Gordon and derivative non-linear Schrödinger hierarchies and some other integrable hierarchies, such as the (w3) non-linear Schrödinger hierarchy and its doubled form.

MSC:
37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)
17B80 Applications of Lie algebras and superalgebras to integrable systems
35Q53 KdV equations (Korteweg-de Vries equations)
37K30 Relations of infinite-dimensional Hamiltonian and Lagrangian dynamical systems with infinite-dimensional Lie algebras and other algebraic structures
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