| 1. |
- Euler, Marianna, et al.
(author)
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Linearizable hierarchies of evolution equations in (1+1) dimensions
- 2003
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In: Studies in applied mathematics (Cambridge). - : Wiley. - 0022-2526 .- 1467-9590. ; 111:3, s. 315-337
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Journal article (peer-reviewed)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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| 2. |
- Euler, Marianna, et al.
(author)
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n-Dimensional real wave equations and the D’Alembert-Hamilton system
- 2001
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In: Nonlinear Analysis. - 0362-546X .- 1873-5215. ; 47:8, s. 5125-5133
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Journal article (peer-reviewed)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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| 3. |
- Euler, Norbert, et al.
(author)
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A tree of linearisable second-order evolution equations by generalised hodograph transformations
- 2001
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In: Journal of Nonlinear Mathematical Physics. - : Springer Science and Business Media LLC. - 1402-9251 .- 1776-0852. ; 8:3, s. 342-362
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Journal article (peer-reviewed)abstract
- We present a list of (1 + 1)-dimensional second-order evolution equations all connected via a proposed generalised hodograph transformation, resulting in a tree of equations transformable to the linear second-order autonomous evolution equation. The list includes autonomous and nonautonomous equations
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| 4. |
- Euler, Norbert, et al.
(author)
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Auto-hodograph transformations for a hierarchy of nonlinear evolution equations
- 2001
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In: Journal of Mathematical Analysis and Applications. - : Elsevier BV. - 0022-247X .- 1096-0813. ; 257:1, s. 21-28
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Journal article (peer-reviewed)abstract
- We introduce nonlocal auto-hodograph transformations for a hierarchy of nonlinear evolution equations. This is accomplished by composing nonlocal transformations (one of which is a hodograph transformation) which linearize the given equations. This enables one to construct sequences of exact solutions for any equation belonging to the hierarchy.
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| 5. |
- Euler, Norbert, et al.
(author)
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Classes of integrable autonomous evolution equations by generalized hodograph transformations
- 2000
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Reports (other academic/artistic)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.
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| 6. |
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| 7. |
- Euler, Norbert, et al.
(author)
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Sundman symmetries of nonlinear second-order and third-order ordinary differential equations
- 2004
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In: Journal of Nonlinear Mathematical Physics. - : Springer Science and Business Media LLC. - 1402-9251 .- 1776-0852. ; 11:3, s. 399-421
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Journal article (peer-reviewed)abstract
- Sundman symmetries arise from more general transformations than do point or contact symmetries. This paper first shows how to systematically calculate Sundman symmetries of second- and third-order nonlinear ordinary differential equations. Secondly, the authors illustrate the application of these symmetries by computing first integrals of the corresponding equations.
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| 8. |
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| 9. |
- Petersson, Niclas, et al.
(author)
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Recursion operators for a class of integrable third-order evolution equations
- 2004
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In: Studies in applied mathematics (Cambridge). - : Wiley. - 0022-2526 .- 1467-9590. ; 112:2, s. 201-225
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Journal article (peer-reviewed)abstract
- We consider ut=uuxxx+n(u)uxuxx+m(u)u3x+r(u)uxx+p(u)u2x+q(u)ux+s(u) with α= 0 and α= 3, for those functional forms of m, n, p, q, r, s for which the equation is integrable in the sense of an infinite number of Lie–Bäcklund symmetries. Recursion operators which are x- and t-independent that generate these infinite sets of (local) symmetries are obtained for the equations. A combination of potential forms, hodograph transformations, and x-generalized hodograph transformations are applied to the obtained equations.
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