We investigate the Sundman symmetries of second-order and third-order nonlinear ordinary differential equations. These symmetries, which are in general nonlocal transformations, arise from generalised Sundman transformations of autonomous equations. We show that these transformations and symmetries can be calculated systematically and can be used to find first integrals of the equations.
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.
Abstract:In our article [5], "A tree of linearisable second-order evolution equations by generalised hodograph transformations [J. Nonlin. Math. Phys. 8 (2001), 342-362] we presented a tree of linearisable (C-integrable) second-order evolution equations in (1+1) dimensions. Expanding this result we report here the complete set of recursion operators for this tree and present several linearisable (C-integrable) hierarchies in (1+1) dimensions.Subject Classification (AMS 2000): 37K35, 37K10, 35Q58.
We consider u t = u α u xxx +n(u)u x u xx +m(u)ux +q(u)u x +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. Local xand t-independent recursion operators that generate these infinite sets of symmetries are obtained for the equations. A combination of potential forms, hodograph transformations and x-generalised hodograph transformations are applied to the obtained equations.
We calculate in detail the conditions which allow the most general third order ordinary differential equation to be linearised in X (T) = 0 under the transformation X(T) = F (x, t), dT = G(x, t)dt. Further generalisations are considered.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.