| 1. |
- Ibragimov, Nail, et al.
(författare)
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Three-dimensional dynamical systems admitting nonlinear superposition with three-dimensional Vessiot-Guldberg-Lie algebras
- 2016
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Ingår i: Applied Mathematics Letters. - : Elsevier. - 0893-9659 .- 1873-5452. ; 52, s. 126-131
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Tidskriftsartikel (refereegranskat)abstract
- The recent method of integration of non-stationary dynamical systems admitting nonlinear superpositions is applied to the three-dimensional dynamical systems associated with three-dimensional Vessiot-Guldberg-Lie algebras L-3. The investigation is based on Bianchi's classification of real three-dimensional Lie algebras and realizations of these algebras in the three-dimensional space. Enumeration of the Vessiot-Guldberg-Lie algebras L-3 allows to classify three-dimensional dynamical systems admitting nonlinear superpositions into thirty one standard types by introducing canonical variables. Twenty four of them are associated with solvable Vessiot-Guldberg-Lie algebras and can be reduced to systems of first-order linear equations. The remaining seven standard types are nonlinear. Integration of the latter types is an open problem. (C) 2015 Elsevier Ltd. All rights reserved.
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| 2. |
- Anco, S., et al.
(författare)
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Symmetries and conservation laws of the generalized Krichever-Novikov equation
- 2016
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Ingår i: Journal of Physics A. - : Institute of Physics Publishing (IOPP). - 1751-8113 .- 1751-8121. ; 49:10
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Tidskriftsartikel (refereegranskat)abstract
- A computational classification of contact symmetries and higher-order local symmetries that do not commute with t, x, as well as local conserved densities that are not invariant under t, x is carried out for a generalized version of the Krichever-Novikov (KN) equation. Several new results are obtained. First, the KN equation is explicitly shown to have a local conserved density that contains t, x. Second, apart from the dilational point symmetries known for special cases of the KN equation and its generalized version, no other local symmetries with low differential order are found to contain t, x. Third, the basic Hamiltonian structure of the KN equation is used to map the local conserved density containing t, x into a nonlocal symmetry that contains t, x. Fourth, a recursion operator is applied to this nonlocal symmetry to produce a hierarchy of nonlocal symmetries that have explicit dependence on t, x. When the inverse of the Hamiltonian map is applied to this hierarchy, only trivial conserved densities are obtained.
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| 3. |
- Gainetdinova, A., et al.
(författare)
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Group classification of ODE y‴ = F (x, y, y′)
- 2014
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Ingår i: Communications in nonlinear science & numerical simulation. - : Elsevier. - 1007-5704 .- 1878-7274. ; 19:2, s. 345-349
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Tidskriftsartikel (refereegranskat)abstract
- In his extensive work of 1884 on the group classification of ordinary differential equations Lie performed, inter alia, the group classification of the particular type of the second-order equations y″ = F (x, y). In the present paper we extend Lie's classification to the third-order equations y‴ = F (x, y, y′).
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| 4. |
- Ibragimov, Nail, et al.
(författare)
-
Classification and integration of four-dimensional dynamical systems admitting non-linear superposition
- 2017
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Ingår i: International Journal of Non-Linear Mechanics. - : Elsevier. - 0020-7462 .- 1878-5638. ; 90, s. 50-71
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Tidskriftsartikel (refereegranskat)abstract
- The method of integration of dynamical systems admitting non-linear superpositions is applied to four-dimensional non-linear dynamical systems. All four-dimensional dynamical systems admitting non-linear superpositions with four-dimensional Vessiot-Guldberg-Lie algebras are classified into 160 standard forms. The integration method is described and illustrated.
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