| 1. |
- Calogero, F., et al.
(author)
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New evolution PDEs with many isochronous solutions
- 2009
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In: Journal of Mathematical Analysis and Applications. - : Elsevier BV. - 0022-247X .- 1096-0813. ; 353:2, s. 481-488
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Journal article (peer-reviewed)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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| 2. |
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| 3. |
- Euler, Marianna, et al.
(author)
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Reciprocal Bäcklund transformations of autonomous evolution equations
- 2009
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In: Theoretical and mathematical physics. - : Springer Science and Business Media LLC. - 0040-5779 .- 1573-9333. ; 159:3, s. 770-778
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Journal article (peer-reviewed)abstract
- We discuss the construction of reciprocal Bäcklund transformations for evolution equations using integrating factors of zeroth and higher orders with their corresponding conservation laws. As an example, we consider the Harry Dym equation and the Schwarzian KdV equation.
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| 4. |
- Euler, Marianna, et al.
(author)
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Second-order recursion operators of third-order evolution equations with fourth-order integrating factors
- 2007
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In: Journal of Nonlinear Mathematical Physics. - : Springer Science and Business Media LLC. - 1402-9251 .- 1776-0852. ; 14:3, s. 321-323
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Journal article (peer-reviewed)abstract
- We report the recursion operators for a class of symmetry integrable evolution equa- tions of third order which admit a fourth-order integrating factor. Under some as- sumptions we obtain the complete list of equations, one of which is a special case of the Schwarzian Korteweg-de Vries equation.
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| 5. |
- Euler, Marianna, et al.
(author)
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The Riccati and Ermakov-Pinney hierarchies
- 2007
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In: Journal of Nonlinear Mathematical Physics. - : Springer Science and Business Media LLC. - 1402-9251 .- 1776-0852. ; 14:2, s. 290-302
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Journal article (peer-reviewed)abstract
- The concept and use of recursion operators is well-established in the study of evolution, in particular nonlinear, equations. We demonstrate the application of the idea of recursion operators to ordinary differential equations. For the purposes of our demonstration we use two equations, one chosen from the class of linearisable hierarchies of evolution equations studied by Euler et al (Stud Appl Math 111 (2003) 315-337) and the other from the class of integrable but nonlinearisible equations studied by Petersson et al (Stud Appl Math 112 (2004) 201-225). We construct the hierarchies for each equation. The symmetry properties of the first hierarchy are considered in some detail. For both hierarchies we apply the singularity analysis. For both we observe intersting behaviour of the resonances for the different possible leading order behaviours. In particular we note the proliferation of subsidiary solutions as one ascends the hierarchy.
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| 6. |
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| 7. |
- Euler, Norbert, et al.
(author)
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Multipotentialisations and iterating-solution formulae : the Krichever-Novikov equation
- 2009
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In: Journal of Nonlinear Mathematical Physics. - 1402-9251 .- 1776-0852. ; 16:Suppl. 1, s. 93-106
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Journal article (peer-reviewed)abstract
- We derive solution-formulae for the Krichever-Novikov equation by a systematic multipotentialisation of the equation. The formulae are achieved due to the connections of the Krichever-Novikov equations to certain symmetry-integrable 3rd-order evolution equations which admit autopotentialisations.
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| 8. |
- Euler, Norbert, et al.
(author)
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On nonlocal symmetries, nonlocal conservation laws and nonlocal transformations of evolution equations : two linearisable hierarchies
- 2009
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In: Journal of Nonlinear Mathematical Physics. - 1402-9251 .- 1776-0852. ; 16:4, s. 489-504
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Journal article (peer-reviewed)abstract
- We discuss nonlocal symmetries and nonlocal conservation laws that follow from the systematic potentialisation of evolution equations. Those are the Lie point symmetries of the auxiliary systems, also known as potential symmetries.We define higher-degree potential symmetries which then lead to nonlocal conservation laws and nonlocal transformations for the equations. We demonstrate our approach and derive second degree potential symmetries for the Burgers' hierarchy and the Calogero-Degasperis-Ibragimov-Shabat hierarchy.
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| 9. |
- Euler, Norbert, et al.
(author)
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Aspects of proper differential sequences of ordinary differential equations
- 2009
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In: Theoretical and mathematical physics. - : Springer Science and Business Media LLC. - 0040-5779 .- 1573-9333. ; 159:1, s. 474-487
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Journal article (peer-reviewed)abstract
- We define a proper differential sequence of ordinary differential equations and introduce a method for deriving an alternate sequence of integrals for such a sequence. We describe some general properties, illustrated by several examples.
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| 10. |
- Leach, Peter, et al.
(author)
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A novel Riccati sequence
- 2009
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In: Journal of Nonlinear Mathematical Physics. - 1402-9251 .- 1776-0852. ; 16:Suppl. 1, s. 157-164
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Journal article (peer-reviewed)abstract
- Hierarchies of evolution partial differential equations have become well-established in the literature over the last thirty years. More recently sequences of ordinary differential equations have been introduced. Of these perhaps the most notable is the Riccati Sequence which has beautiful singularity, symmetry and integrability properties. We examine a variation of this sequence and find that there are some remarkable changes in properties consequential upon this variation.
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