| 1. |
- Constantin, Adrian
(author)
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A Hamiltonian formulation for free surface water waves with non-vanishing vorticity
- 2005
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In: Journal of Nonlinear Mathematical Physics. - : Springer Science and Business Media LLC. - 1402-9251 .- 1776-0852. ; 12:Suppl. 1, s. 202-211
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Journal article (peer-reviewed)abstract
- We describe the derivation of a formalism in the context of the governing equations for two-dimensional water waves propagating over a flat bed in a flow with non-vanishing vorticity. This consists in providing a Hamiltonian structure in terms of two variables which are scalar functions.
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| 2. |
- Ehrnström, Mats
(author)
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A note on surface profiles for symmetric gravity waves with vorticity
- 2006
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In: Journal of Nonlinear Mathematical Physics. - : Springer Science and Business Media LLC. - 1402-9251 .- 1776-0852. ; 13:1, s. 1-8
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Journal article (peer-reviewed)abstract
- We consider a nontrivial symmetric periodic gravity wave on a current with nondecreasing vorticity. It is shown that if the surface profile is monotone between trough and crest, it is in fact strictly monotone. The result is valid for both finite and infinite depth.
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| 3. |
- Ehrnström, Mats, et al.
(author)
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On the fluid motion in standing waves
- 2008
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In: Journal of Nonlinear Mathematical Physics. - : Springer Science and Business Media LLC. - 1402-9251 .- 1776-0852. ; 15:suppl 2, s. 74-86
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Journal article (peer-reviewed)abstract
- This paper concerns linear standing gravity water waves on finite depth. We obtain qualitative and quantitative understanding of the particle paths within the wave.
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| 4. |
- Ehrnström, Mats
(author)
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Uniqueness of steady symmetric deep-water waves with vorticity
- 2005
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In: Journal of Nonlinear Mathematical Physics. - : Springer Science and Business Media LLC. - 1402-9251 .- 1776-0852. ; 12:1, s. 27-30
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Journal article (peer-reviewed)abstract
- Given a steady and symmetric deep-water wave we prove that the surface profile and the vorticity distribution determine the wave motion completely throughout the fluid.
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| 5. |
- Euler, Marianna
(author)
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Fourth-order recursion operators for third-order evolution equations
- 2008
-
In: Journal of Nonlinear Mathematical Physics. - : Springer Science and Business Media LLC. - 1402-9251 .- 1776-0852. ; 15:2, s. 147-151
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Journal article (peer-reviewed)abstract
- We report the recursion operators for a class of symmetry integrable evolution equations of third order which admit fourth-order recursion operators. Under the given assumptions we obtain the complete list of equations, one of which is the well-known Krichever-Novikov equation.
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| 6. |
- 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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| 7. |
- 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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| 8. |
- 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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| 9. |
- 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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| 10. |
- Goodall, R., et al.
(author)
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Generalised symmetries and the ermakov-lewis invariant
- 2005
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In: Journal of Nonlinear Mathematical Physics. - : Atlantis Press. - 1402-9251 .- 1776-0852. ; 12:1, s. 15-26
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Journal article (other academic/artistic)abstract
- Generalised symmetries and point symmetries coincide for systems of first-order ordinary differential equations and are infinite in number. Systems of linear first-order ordinary differential equations possess a generalised rescaling symmetry. For the system of first-order ordinary differential equations corresponding to the time-dependent linear oscillator the invariant of this symmetry has the form of the famous Ermakov-Lewis invariant, but in fact reveals a richer structure. © 2005 Taylor & Francis Group, LLC.
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