| 1. |
- 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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| 2. |
- Ibragimov, Nail, et al.
(författare)
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Conservation laws and solutions of a quantum drift-diffusion model for semiconductors
- 2015
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Ingår i: International Journal of Non-Linear Mechanics. - : Elsevier. - 0020-7462 .- 1878-5638. ; 77, s. 69-73
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Tidskriftsartikel (refereegranskat)abstract
- A non-linear system of partial differential equations describing a quantum drift-diffusion model for semiconductor devices is investigated by methods of group analysis. An infinite number of conservation laws associated with symmetries of the model are found. These conservation laws are used for representing the system of equations under consideration in the conservation form. Exact solutions provided by the method of conservation laws are discussed. These solutions are different from invariant solutions. (C) 2015 Published by Elsevier Ltd.
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| 3. |
- Ibragimov, Nail, et al.
(författare)
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Group analysis of the drift–diffusion model for quantum semiconductors
- 2015
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Ingår i: Communications in nonlinear science & numerical simulation. - : Elsevier. - 1007-5704 .- 1878-7274. ; 20:1, s. 74-78
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Tidskriftsartikel (refereegranskat)abstract
- In the present paper a quantum drift–diffusion model describing semi-conductor devices is considered. New conservation laws for the model are computed and used to construct exact solutions.
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| 4. |
- Ibragimov, Nail, et al.
(författare)
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Group classification and conservation laws of anisotropic wave equations with a source
- 2016
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Ingår i: Journal of Mathematical Physics. - : American Institute of Physics (AIP). - 0022-2488 .- 1089-7658. ; 57:8
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Tidskriftsartikel (refereegranskat)abstract
- Linear and nonlinear waves in anisotropic media are useful in investigating complex materials in physics, biomechanics, biomedical acoustics, etc. The present paper is devoted to investigation of symmetries and conservation laws for nonlinear anisotropic wave equations with specific external sources when the equations in question are nonlinearly self-adjoint. These equations involve two arbitrary functions. Construction of conservation laws associated with symmetries is based on the generalized conservation theorem for nonlinearly self-adjoint partial differential equations. First we calculate the conservation laws for the basic equation without any restrictions on the arbitrary functions. Then we make the group classification of the basic equation in order to specify all possible values of the arbitrary functions when the equation has additional symmetries and construct the additional conservation laws.
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| 5. |
- Ibragimov, Nail, et al.
(författare)
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Nonlinear self-adjointness, conservation laws, and the construction of solutions of partial differential equations using conservation laws
- 2013
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Ingår i: Russian Mathematical Surveys. - : TURPION LTD, IOP PUBLISHING. - 0036-0279 .- 1468-4829. ; 68:5, s. 889-921
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Tidskriftsartikel (refereegranskat)abstract
- The method of nonlinear self-adjointness, which was recently developed by the first author, gives a generalization of Noether's theorem. This new method significantly extends approaches to constructing conservation laws associated with symmetries, since it does not require the existence of a Lagrangian. In particular, it can be applied to any linear equations and any nonlinear equations that possess at least one local conservation law. The present paper provides a brief survey of results on conservation laws which have been obtained by this method and published mostly in recent preprints of the authors, along with a method for constructing exact solutions of systems of partial differential equations with the use of conservation laws. In most cases the solutions obtained by the method of conservation laws cannot be found as invariant or partially invariant solutions.
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