| 1. |
- Euler, Marianna, et al.
(författare)
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On the construction of approximate solutions for a multidimensional nonlinear heat equation
- 1994
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Ingår i: Journal of Physics A. - : IOP Publishing. - 0305-4470 .- 1361-6447. ; 27:6, s. 2083-2092
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Tidskriftsartikel (refereegranskat)abstract
- Summary: We study three methods, based on continuous symmetries, to find approximate solutions for the multidimensional nonlinear heat equation $\partial u/\partial x_0+ \Delta u= au^n+ \varepsilon f(u)$, where $a$ and $n$ are arbitrary real constants, $f$ is a smooth function, and $0< \varepsilon\ll 1$.
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| 2. |
- Euler, Marianna, et al.
(författare)
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Symmetry classification for a coupled nonlinear Schrödinger equation
- 1994
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Ingår i: Journal of Nonlinear Mathematical Physics. - : Springer Science and Business Media LLC. - 1402-9251 .- 1776-0852. ; 1:4, s. 358-379
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Tidskriftsartikel (refereegranskat)abstract
- We do a Lie symmetry classification for a system of two nonlinear coupled Schrödinger equations. Our system under consideration is a generalization of the equations which follow from the analysis of optical fibres. Reductions of some special equations are given.
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| 3. |
- Euler, Norbert, et al.
(författare)
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Conditional and approximate symmetries for a generalized van der Pol equation
- 1994
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Ingår i: Lie Groups and their Applications. ; 1:1, s. 79-94
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Tidskriftsartikel (refereegranskat)abstract
- Non-Lie ansätze and exact solutions are constructed by means of conditional symmetries for a class of nonlinear heat equations in one space dimension. We study approximate symmetries by which one can obtain approximate solutions for multidimensional partial differential equations. The method of approximate symmetries is applied to an $n$-dimensional generalized van der Pol equation. Approximate conditions symmetries are introduced to construct approximate solutions for the latter equation.
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| 4. |
- Euler, Norbert, et al.
(författare)
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Symmetry properties of the approximations of multidimensional generalized van der Pol equations
- 1994
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Ingår i: Journal of Nonlinear Mathematical Physics. - : Springer Science and Business Media LLC. - 1402-9251 .- 1776-0852. ; 1:1, s. 41-59
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Tidskriftsartikel (refereegranskat)abstract
- The subject of the paper are symmetries of the nonlinear hyperbolic equation, $$\frac{\partial^2u}{\partial t^2} - \sum_{n=1}^N \frac{\partial^2u}{\partial^2x_n} +m^2u - \varepsilon f(u)\left(\lambda_0 \frac{\partial u}{\partial t} + \sum_{n=1}^N\lambda_n \frac{\partial u}{\partial x_n}\right) =0. $$ The case $f(u)=1-u^2$ corresponds to the generalized van der Pol equation. The equation is expanded in powers of the parameter $\varepsilon$, which stands in front of the nonlinear term, and then symmetries of the resulting chain of approximate equations are studied by means of the Lie-group technique. Emphasis is made on a special type of the function $f(u)$ which admits conformal invariance of the equation.
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| 5. |
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| 6. |
- Duarte, L.G.S., et al.
(författare)
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Invertible point transformations, Painlevé analysis and anharmonic oscillators
- 1990
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Ingår i: Journal of Physics A. - : IOP Publishing. - 0305-4470 .- 1361-6447. ; 23, s. 1457-1463
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Tidskriftsartikel (refereegranskat)abstract
- The techniques of an invertible point transformation and the Painleve analysis can be used to construct integrable ordinary differential equations. The authors compare both techniques for anharmonic oscillators.
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| 7. |
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| 8. |
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| 9. |
- Euler, Norbert, et al.
(författare)
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Invertible point transformations, Painlevé test, and the second Painlevé transcendent
- 1991
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Ingår i: International journal of theoretical physics. - 0020-7748 .- 1572-9575. ; 30:9, s. 1267-1271
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Tidskriftsartikel (refereegranskat)abstract
- The techniques of invertible point transformations and the Painlevé analysis can be used to construct integrable ordinary differential equations. We compare both techniques for the second Painlevé transcendent.
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| 10. |
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