| 1. |
- Basarab-Horwath, P., et al.
(författare)
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Amplitude-phase representation for solutions of nonlinear d'Alembert equations
- 1995
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Ingår i: Journal of Physics A. - : IOP Publishing. - 0305-4470 .- 1361-6447. ; 28:21, s. 6193-6201
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Tidskriftsartikel (refereegranskat)abstract
- We consider the nonlinear complex d'Alembert equation Square Operator Psi =F( mod Psi mod ) Psi with Psi represented in terms of amplitude and phase, in (1+n)-dimensional Minkowski space. We exploit a compatible d'Alembert-Hamilton system to construct new types of exact solutions for some nonlinearities.
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| 2. |
- Calogero, F., et al.
(författare)
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New evolution PDEs with many isochronous solutions
- 2009
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Ingår i: Journal of Mathematical Analysis and Applications. - : Elsevier BV. - 0022-247X .- 1096-0813. ; 353:2, s. 481-488
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Tidskriftsartikel (refereegranskat)abstract
- Certain nonlinear evolution PDEs in 1 + 1 variables (time and space) are identified, featuring a positive parameter ω and evolving, for a large class of initial data, periodically with the fixed period T = 2 π / ω (or perhaps over(T, ̃) = p T with p a small integer). They are autonomous (i.e., they do not feature any explicit dependence on the time variable), but they generally (although not quite all of them) depend explicitly on the space variable hence are not translation-invariant. They are integrable, having been obtained by applying an appropriate change of dependent and independent variables to certain nonlinear evolution PDEs whose integrable character has been recently ascertained. Solutions of some of these PDEs are exhibited.
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| 3. |
- Euler, Marianna, et al.
(författare)
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A class of semilinear fifth-order evolution equations : recursion operators and multipotentialisations
- 2011
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Ingår i: Journal of Nonlinear Mathematical Physics. - 1402-9251 .- 1776-0852. ; 18:Suppl. 1, s. 61-75
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Tidskriftsartikel (refereegranskat)abstract
- We apply a list of criteria which leads to a class of fifth-order symmetry-integrable evolution equations. The recursion operators for this class are given explicitly. Multipotentialisations are then applied to the equations in this class in order to extend this class of integrable equations.
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| 4. |
- Euler, Marianna, 1961-, et al.
(författare)
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Aufgaben, Theorie und Lösungen zur Linearen Algebra, Teil 1 : Der Euklidische Raum
- 2017
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Bok (refereegranskat)abstract
- Dieses Buch ist der erste Teil einer dreiteiligen Serie mit dem Titel Aufgaben, Theorieund L¨osungen zur Linearen Algebra. Dieser erste Teil behandelt Vektoren des euklidischen Raumes sowie Matrizen, Matrixalgebra und Systeme von linearen Gleichungen. Wir l¨osen lineare System mit Hilfe des Gaußschen Eliminationsverfahrens und auf anderem Wege und untersuchen die Eigenschaften dieser Systeme in Bezug auf Vektoren und Matrizen. Dar¨uber hinaus betrachten wir lineare Abbildungen und berechnendie Standardmatrizen dieser Abbildungen.
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| 5. |
- Euler, Marianna, et al.
(författare)
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Explicitly space- and time-dependent d'Alembert equations with symmetries
- 1999
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Ingår i: International Journal of Modern Physics A. - 0217-751X .- 1793-656X. ; 14:26, s. 4189-4200
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Tidskriftsartikel (refereegranskat)abstract
- The general d'Alembert equation $\square u+ f(x_0, x_1,u)= 0$ is considered, where $\square$ is the two-dimensional d'Alembert operator. We classify the equation for functions $f$ by which it admits several Lie symmetry algebras, which include the Lorentz symmetry generator. The corresponding symmetry reductions are listed.
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| 6. |
- Euler, Marianna, et al.
(författare)
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Integrating factors and conservation laws for some Camassa-Holm type equations
- 2012
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Ingår i: Communications on Pure and Applied Analysis. - : American Institute of Mathematical Sciences (AIMS). - 1534-0392 .- 1553-5258. ; 11:4, s. 1421-1430
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Tidskriftsartikel (refereegranskat)abstract
- We classify all 1st-order integrating factors and the correspondingconservation laws for a class of Camassa-Holm type equations.
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| 7. |
- Euler, Marianna, et al.
(författare)
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Invariance of the Kaup-Kupershmidt equation and triangular auto-Bäcklund transformations
- 2012
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Ingår i: Journal of Nonlinear Mathematical Physics. - 1402-9251 .- 1776-0852. ; 19:3, s. 1220001/1-1220001/7
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Tidskriftsartikel (refereegranskat)abstract
- We report triangular auto-Bäcklund transformations for the solutions of a fifth-order evolution equation, which is a constraint for an invariance condition of the Kaup-Kupershmidt equation derived by E. G. Reyes in his paper titled ``Nonlocal symmetries and the Kaup-Kupershmidt equation'' [ Math. Phys., 46, 073507, 19 pp., 2005]. These auto-Bäcklund transformations can then be applied to generate solutions of the Kaup-Kupershmidt equation. We show that triangular auto-Bäcklund transformations result from a systematic multipotentialisation of the Kupershmidt equation.
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| 8. |
- Euler, Marianna, et al.
(författare)
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Linearizable hierarchies of evolution equations in (1+1) dimensions
- 2003
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Ingår i: Studies in applied mathematics (Cambridge). - : Wiley. - 0022-2526 .- 1467-9590. ; 111:3, s. 315-337
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Tidskriftsartikel (refereegranskat)abstract
- In our article [1], "A tree of linearisable second-order evolution equations by generalised hodograph transformations" we present a class of linearizable (C-integrable) second-order evolution equations in (1+1) dimensions, using a generalized hodograph transformation. We report here the complete set of recursion operators for this class and present the resulting linearizable (C-integrable) hierarchies in (1+1) dimensions. The autonomous class of linearizable hierarchies are extended further by considering the equations in potential form followed by the pure hodograph transformation.
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| 9. |
- Euler, Marianna, 1961-, et al.
(författare)
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Multipotentializations and nonlocal symmetries: Kupershmidt, Kaup-Kupershmidt and Sawada-Kotera equations
- 2017
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Ingår i: Journal of Nonlinear Mathematical Physics. - : Taylor & Francis. - 1402-9251 .- 1776-0852. ; 24:3, s. 303-314
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Tidskriftsartikel (övrigt vetenskapligt/konstnärligt)abstract
- In this letter we report a new invariant for the Sawada-Kotera equation that is obtained by a systematic potentialization of the Kupershmidt equation. We show that this result can be derived from nonlocal symmetriesand that, conversely, a previously known invariant of the Kaup-Kupershmidt equation can be recovered using potentializations.
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| 10. |
- Euler, Marianna, et al.
(författare)
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n-Dimensional real wave equations and the D’Alembert-Hamilton system
- 2001
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Ingår i: Nonlinear Analysis. - 0362-546X .- 1873-5215. ; 47:8, s. 5125-5133
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Tidskriftsartikel (refereegranskat)abstract
- We reduce the nonlinear wave equation □nu = αF[exp(βu)] to ordinary differential equtions and construct exact solutions, by the use of a compatible d'Alembert-Hamilton system. The solutions of these ordinary differential equations, together with the solutions of the corresponding d'Alembert-Hamilton equations, provide a rich class of exact solutions of the multidimensional wave equations. The wave equations are studied in n-dimensional Minkowski space.
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