| 1. |
- 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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| 2. |
- Ibragimov, R. N., et al.
(författare)
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Conservation laws and invariant solutions of the non-linear governing equations associated with a thermodynamic model of a rotating detonation engines with Korobeinikov's chemical source term
- 2016
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Ingår i: International Journal of Non-Linear Mechanics. - : Elsevier. - 0020-7462 .- 1878-5638. ; 78, s. 29-34
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Tidskriftsartikel (refereegranskat)abstract
- The non-linear governing gas dynamics equations that are used as a descriptor of a rotating detonation engine are investigated from the group theoretical standpoint. The equations incorporate approximation of Korobeinikov's chemical reaction model that are used to describe the two-dimensional detonation field on a surface of a two-dimensional cylindrical chamber without thickness. The transformations that leave the equations invariant are found. On the basis of these transformations, the conservation equations were constructed and the invariant solutions were obtained for specific form of the equation of state, for which the equations are non-linearly self-adjoint. The invariant solutions are given in terms of the functions that satisfy non-linear ordinary differential equations. The above reduction simplifies the analysis of the original non-linear system of partial differential equations on a surface of rotating cylinder. (C) 2015 Elsevier Ltd. All rights reserved.
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| 3. |
- 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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| 4. |
- Galiakberova, L. R., et al.
(författare)
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Nonlinear self-adjointness of the Krichever-Novikov equation
- 2014
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Ingår i: Communications in nonlinear science & numerical simulation. - : Elsevier. - 1007-5704 .- 1878-7274. ; 19:2, s. 361-363
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Tidskriftsartikel (refereegranskat)abstract
- It is known that the classification of third-order evolutionary equations with the constant separant possessing a nontrivial Lie-Bäcklund algebra (in other words, integrable equations) results in the linear equation, the KdV equation and the Krichever-Novikov equation. The first two of these equations are nonlinearly self-adjoint. This property allows to associate conservation laws of the equations in question with their symmetries. The problem on nonlinear self-adjointness of the Krichever-Novikov equation has not been solved yet. In the present paper we solve this problem and find the explicit form of the differential substitution providing the nonlinear self-adjointness.
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| 5. |
- Ibragimov, Nail, 1939-, et al.
(författare)
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Chaplygin gas motions associated with nonlocal conservation laws
- 2017
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Ingår i: JOURNAL OF COUPLED SYSTEMS AND MULTISCALE DYNAMICS. - : AMER SCIENTIFIC PUBLISHERS. - 2330-152X. ; 5:2-4, s. 63-68
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Tidskriftsartikel (refereegranskat)abstract
- The recent method of conservation laws for constructing exact solutions of partial differential equations is applied to the nonlocal conservation laws of the Chaplygin gas. The nonlocal conservation laws provide twenty different types of exact solutions. They are listed in three tables. Seven types of these solutions describe isentropic flows satisfying Chaplygin's relation between the pressure and density. All solutions are written in the explicit form and contain either arbitrary functions or arbitrary constants.
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| 6. |
- 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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| 7. |
- 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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| 8. |
- Ibragimov, Nail H., et al.
(författare)
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Symmetries and Conservation Laws of a Spectral Nonlinear Model for Atmospheric Baroclinic Jets
- 2014
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Ingår i: Mathematical Modelling of Natural Phenomena. - : EDP SCIENCES S A.. - 0973-5348 .- 1760-6101. ; 9:5, s. 111-118
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Tidskriftsartikel (refereegranskat)abstract
- In this paper, we shall obtain the symmetries of the mathematical model describing spontaneous relaxation of eastward jets into a meandering state and use these symmetries for constructing the conservation laws. The basic eastward jet is a spectral parameter of the model, which is in geostrophic equilibrium with the basic density structure and which guarantees the existence of nontrivial conservation laws.
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