# zbMATH — the first resource for mathematics

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
Full Text:
##### References:
 [1] Zaharov, V.E.; Shabat, A.B., Integration of nonlinear equations of mathematical physics using the method of inverse scattering problem, Funct. anal. appl., 13, 3, 13-21, (1979) [2] Tahtadjan, L.A.; Faddejev, L.D., Hamiltonian approach in the theory of solitons, (1986), Nauka Moscow [3] A.C. Newell, Solitons in mathematics and physics, University of Arizona Society for Industrial and Applied mathematics, 1985 · Zbl 0565.35003 [4] Flaschka, H.; Newell, A.C.; Ratiu, T., Kac – moody Lie algebras and soliton equations, Physica D, 9, 303-323, (1983), 324-332 · Zbl 0643.35098 [5] P.I. Holod, Integrable hamiltonian systems on the orbits of affine Lie groups and periodical problem for mKdV equation, Kiev preprint, ITF-82-144R, 1982 [6] Reyman, A.G.; Semenov Tian-Shansky, M.A., Group theoretical methods in the theory of finite-dimensional integrable systems, VINITI: cont. probl. math. fundamental trends, 6, 145-147, (1989) [7] Holod, P.I., Hamiltonian systems connected with the anisotropic affine Lie algebras and higher landau – lifschitz equations, Dokl. acad. of sciences of Ukrainian SSR, 276, 5, 5-8, (1984) [8] Holod, P.I., Hidden symmetry of landau – lifschitz equations, its higher analogues and dual equation for asymmetric chiral field, Theoret. and math. phys., 70, 1, 18-29, (1987) [9] Holod, P.I.; Skrypnyk, T.V., Anisotropic quasigraded Lie algebras on the algebraic curves and integrable Hamiltonian systems, Naukovi zapysky NAUKMA, ser phys-math sciences, 18, 20-25, (2000), (in Ukrainian) · Zbl 0974.17036 [10] T.V. Skrypnyk, Lie algebras on algebraic curves and finite-dimensional integrable systems, in: Proceedings of the XXIII International Colloquium on the Group Theoretical Methods in Physics held in Dubna (Russia), 1-5 August 2000. e-print, nlin.SI-0010005 [11] Skrypnyk, T.V., Quasigraded Lie algebras on hyperelliptic curves and classical integrable systems, J. math. phys., 42, 9, 4570-4581, (2001) · Zbl 1032.17052 [12] Skrypnyk, T.V.; Holod, P.I., Quasigraded Lie algebras on hierarchies of integrable equations, J. phys. A, 34, 9, 1123-1137, (2001) [13] Skrypnyk, T.V., Quasigraded deformations of Lie algebras and finite-dimensional integrable systems, Czech. J. phys., 52, 11, 1283-1288, (2002) · Zbl 1047.37052 [14] Skrypnyk, T.V., Quasigraded Lie algebras and hierarchies of integrable equations, Czech. J. phys., 53, 11, 1119-1124, (2003) [15] Skrypnyk, T.V., Deformations of loop algebras and integrable systems: hierarchies of integrable equations, J. math. phys., 45, 12, 4578-4595, (2004) · Zbl 1064.37055 [16] Golubchik, I.Z.; Sokolov, V.V., Yet another kind of the classical yang – baxter equation, Funktsional. anal. i prilozhen., 34, 4, 75-78, (2000) [17] Golubchik, I.Z.; Sokolov, V.V., Compatible Lie brackets and integrable equations of the principle chiral model type, Funktsional. anal. i prilozhen., 36, 3, 9-19, (2002) · Zbl 1022.17024 [18] Mikhailov, A.V., The reduction problem and the inverse scattering method in the soliton theory, Physica D, 3, 73-117, (1981) · Zbl 1194.37113 [19] de Groot, M.F.; Hollowood, T.J.; Miramontes, J.L., Generalized drinfeld – sokolov hierarchies, Comm. math. phys., 145, 1, 57-85, (1992) · Zbl 0749.35044 [20] Kac, V.G., Infinite-dimensional Lie algebras, (1993), Moscow Mir · Zbl 0574.17002 [21] Drinfeld, V.G.; Sokolov, V.V., Lie algebras and equations of the KdV-type, J. sov. math., 30, (1985) [22] Holod, P.I., Hamiltonian systems on the orbits of affine Lie groups and integrable equations, Physica mnogochastichnyh sistem, 7, 30-39, (1985) [23] Calogero, F.; Degasperis, A., Spectral transformation and solitons, (1985), Mir Moscow [24] Mikhailov, A.V.; Shabat, A.B.; Yamilov, R.I., Symmetry approach to the classification of non-linear integrable equations, Uspekhi mat. nauk, 42, 4, 3-53, (1987) [25] Chen, H., Phys. rev. lett., 33, 15, 925-930, (1974) [26] Borisov, A.B.; Zykov, S.A., Theoret. and math. phys., 115, 2, 199-214, (1998) [27] Zhiber, A.V.; Sokolov, V.V., New example of hyperbolical equations possessing integrals, Theoret. and math. phys., 120, 1, 21-26, (1999) · Zbl 0957.35092 [28] Krichiver, I.M.; Novikov, S.P., Virasoro-type algebras, Riemannian surfaces and structures of the soliton theory, Funct. anal. appl., 21, 2, 46-64, (1987) [29] I.L. Cantor, D.E. Persits, Closed stacks of Poisson brackets, in: Proceedings of the IX USSR Conference in Geometry, Kishinev, Shtinitsa, 1988, p. 141
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.