| 1. |
- 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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| 2. |
- 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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| 3. |
- 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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| 4. |
- Euler, Marianna, et al.
(författare)
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Properties of the Calogero-Degasperis-Ibragimov-Shabat differential sequence
- 2011
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Ingår i: Lobachevskii Journal of Mathematics. - 1995-0802 .- 1818-9962. ; 32:1, s. 61-70
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Tidskriftsartikel (refereegranskat)abstract
- We present a differential sequence based upon the Calogero-Degasperis-Ibragimov-Shabat Equation and determine first integrals and the general solution. Under suitable transformations eachmember of the differential sequence can be recast as a product of two factors and we report some of the properties of the factored form.
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| 5. |
- Euler, Marianna, et al.
(författare)
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The Two-Component Camassa–Holm Equations CH(2,1) and CH(2,2): First-Order Integrating Factors and Conservation Laws
- 2012
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Ingår i: Journal of Nonlinear Mathematical Physics. - 1402-9251 .- 1776-0852. ; 19:Supplement 1, s. 13-22
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Tidskriftsartikel (refereegranskat)abstract
- Recently, Holm and Ivanov, proposed and studied a class of multi-component generalisations of the Camassa-Holm equations [D D Holm and R I Ivanov, Multi-component generalizations of the CH equation: geometrical aspects, peakons and numerical examples, [J. Phys A: Math. Theor, vol. 43, 492001 (20pp), 2010]. We consider two of those systems, denoted by Holm and Ivanov by CH(2,1) and CH(2,2), and report a class of integrating factors and its corresponding conservation laws for these two systems. In particular, we obtain the complete sent of first-order integrating factors for the systems in Cauchy-Kovalevskaya form and evaluate the corresponding sets of conservation laws for CH(2,1) and CH(2,2).
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| 6. |
- Euler, Norbert, et al.
(författare)
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An alternate view on symmetries of second-order linearisable ordinary differential equations
- 2012
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Ingår i: Lobachevskii Journal of Mathematics. - 1995-0802 .- 1818-9962. ; 33:2, s. 191-194
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Tidskriftsartikel (refereegranskat)abstract
- We discuss the Sundman symmetries and point symmetries of linearisable second-order ordinary differential equations, which include discrete and nonlocal symmetries. The method of construction of such symmetries is also introduced for general n-th order equations.
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| 7. |
- Euler, Norbert, et al.
(författare)
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The converse problem for the multipotentialisation of evolution equations and systems
- 2011
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Ingår i: Journal of Nonlinear Mathematical Physics. - 1402-9251 .- 1776-0852. ; 18:Suppl. 1, s. 77-105
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Tidskriftsartikel (refereegranskat)abstract
- We propose a method to identify and classify evolution equations and systems that can be multipotentialised in given target equations or target systems. We refer to this as the converse problem. Although we mainly study a method for (1 + 1)-dimensional equations/system, we do also propose an extension of the methodology to higher-dimensional evolution equations. An important point is that the proposed converse method allows one to identify certain types of auto-Bäcklund transformations for the equations/systems. In this respect we define the triangular-auto-Bäcklund transformation and derive its connections to the converse problem. Several explicit examples are given. In particular, we investigate a class of linearisable third-order evolution equations, a fifth-order symmetry-integrable evolution equation as well as linearisable systems.
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| 8. |
- Euler, Norbert
(författare)
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Linear operators and the general solution of elementary linear ordinary differential equations
- 2012
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Ingår i: CODEE Journal. - : Claremont Colleges Library. - 2160-5211. ; CJ12-1802
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Tidskriftsartikel (refereegranskat)abstract
- We make use of linear operators to derive the formulae for the generalsolution of elementary linear scalar ordinary differential equations of order n.The key lies in the factorization of the linear operators in terms of first-orderoperators. These first-order operators are then integrated by applying theircorresponding integral operators.This leads to the solution formulae for bothhomogeneous- and nonhomogeneous linear differential equations in a naturalway without the need for any ansatz (or “educated guess”). For second-orderlinear equations with nonconstant coefficients, the condition of the factorizationis given in terms of Riccati equations.
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| 9. |
- Leach, Peter, et al.
(författare)
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Symmetries, integrals and solutions of ordinary differential equations of maximal symmetry
- 2010
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Ingår i: Proceedings of the Indian Academy of sciences. Mathematical sciences. - : Springer Science and Business Media LLC. - 0253-4142 .- 0973-7685. ; 120:1, s. 113-130
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Tidskriftsartikel (refereegranskat)abstract
- Second- and third-order scalar ordinary differential equations of maximal symmetry in the traditional sense of point, repsectively contact, symmetries are examined for the mappings they produced in solutions and fundamental first integrals. The properties of ....
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