| 1. |
- Euler, Norbert, et al.
(författare)
-
Classes of integrable autonomous evolution equations by generalized hodograph transformations
- 2000
-
Rapport (övrigt vetenskapligt/konstnärligt)abstract
- We propose a generalized hodograph transformation in order to conduct a classification of integrable nonlinear evolution equations. Classes of both second- and third-order autonomous evolution equations in one dependent and two independent variables are presented. Transformations are derived which transform the equations either to linear or other integrable equations.
|
|
| 2. |
- Euler, Norbert, 1964-, et al.
(författare)
-
Nonlocal invariance of the multipotentialisations of the Kupershmidt equation and its higher-order hierarchies
- 2018. - 1
-
Ingår i: Nonlinear Systems and Their Remarkable Mathematical Structures. - Boca Raton, USA : CRC Press. - 9781138601000 - 9780429470462 ; , s. 317-351
-
Bokkapitel (refereegranskat)abstract
- The term multipotentialisation of evolution equations in 1+1 dimensions refers to the process of potentialising a given evolution equation, followed by at least one further potentialisation of the resulting potential equation. For certain equations this process can be applied several times to result in a finite chain of potential equations, where each equation in the chain is a potential equation of the previous equation. By a potentialisation of an equation with dependent variable u to an equation with dependent variable v, we mean a differential substitution v_x=\Phi^t, where \Phi^t is a conserved current of the equation in u. The process of multipotentialisation may lead to interesting nonlocal transformations between the equations. Remarkably, this can, in some cases, result in nonlocal invariance transformations for the equations, which then serve as iteration formulas by which solutions can be generated for all the equations in the chain.In the current paper we give a comprehensive introduction to this subject and report new nonlocal invariance transformations that result from the multipotentialisation of the Kupershmidt equation and its higher-order hierarchies. The recursion operators that define the hierarchies are given explicitly.
|
|
| 3. |
- Basarab-Horwath, P., et al.
(författare)
-
Amplitude-phase representation for solutions of nonlinear d'Alembert equations
- 1995
-
Ingår i: Journal of Physics A. - : IOP Publishing. - 0305-4470 .- 1361-6447. ; 28:21, s. 6193-6201
-
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.
|
|
| 4. |
- Calogero, F., et al.
(författare)
-
New evolution PDEs with many isochronous solutions
- 2009
-
Ingår i: Journal of Mathematical Analysis and Applications. - : Elsevier BV. - 0022-247X .- 1096-0813. ; 353:2, s. 481-488
-
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.
|
|
| 5. |
|
|
| 6. |
- Duarte, L.G.S., et al.
(författare)
-
Invertible point transformations, Painlevé analysis and anharmonic oscillators
- 1990
-
Ingår i: Journal of Physics A. - : IOP Publishing. - 0305-4470 .- 1361-6447. ; 23, s. 1457-1463
-
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.
|
|
| 7. |
- Euler, Marianna, et al.
(författare)
-
A class of semilinear fifth-order evolution equations : recursion operators and multipotentialisations
- 2011
-
Ingår i: Journal of Nonlinear Mathematical Physics. - 1402-9251 .- 1776-0852. ; 18:Suppl. 1, s. 61-75
-
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.
|
|
| 8. |
- Euler, Marianna, 1961-, et al.
(författare)
-
Aufgaben, Theorie und Lösungen zur Linearen Algebra, Teil 1 : Der Euklidische Raum
- 2017
-
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.
|
|
| 9. |
- Euler, Marianna, et al.
(författare)
-
Explicitly space- and time-dependent d'Alembert equations with symmetries
- 1999
-
Ingår i: International Journal of Modern Physics A. - 0217-751X .- 1793-656X. ; 14:26, s. 4189-4200
-
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.
|
|
| 10. |
- Euler, Marianna, et al.
(författare)
-
Integrating factors and conservation laws for some Camassa-Holm type equations
- 2012
-
Ingår i: Communications on Pure and Applied Analysis. - : American Institute of Mathematical Sciences (AIMS). - 1534-0392 .- 1553-5258. ; 11:4, s. 1421-1430
-
Tidskriftsartikel (refereegranskat)abstract
- We classify all 1st-order integrating factors and the correspondingconservation laws for a class of Camassa-Holm type equations.
|
|
