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The Riemann and Einstein-Weyl geometries in the theory of ordinary differential equations, their applications and all that. (English) Zbl 1066.34004
Shabat, A.B.(ed.) et al., New trends in integrability and partial solvability. Proceedings of the NATO Advanced Research Workshop, Cadiz, Spain, June 12–16, 2002. Dordrecht: Kluwer Academic Publishers (ISBN 1-4020-1835-5/hbk). NATO Science Series II: Mathematics, Physics and Chemistry 132, 115-156 (2004).
Summary: Some properties of the 4-dimensional Riemannian spaces with metrics \[ ds^2= 2(za_3- ta_4) dx^2+ 4(za_2- ta_3) dx\,dy+ 2(za_1- ta_2) dy^2+ 2dx\,dz+ 2dy\,dt \] associated with second-order nonlinear differential equations \[ y''+ a_1(x, y)y^{\prime 3}+ 3a_2(x, y)y^{\prime 2}+ 3a_3(x,y) y'+ a_4(x,y)= 0 \] with arbitrary coefficients \(a_i(x, y)\) are considered. Three-dimensional Einstein-Weyl spaces connected with dual equations \[ b''= g(a,b,b'), \] where the function \(g(a, b, b')\) satisfies the partial differential equation \[ \begin{split} g_{aacc}+ 2cg_{abcc}+ 2gg_{accc}+ c^2 g_{bbcc}+ 2cgg_{bccc}+ g^2 g_{cccc}+ (g_a+ cg_b) g_{ccc}- 4g_{abc}\\ -4cg_{bbc}- cg_c g_{bcc}- 3gg_{bcc}- g_c g_{acc}+ 4g_c g_{bc}- 3g_b g_{cc}+ 6g_{bb}= 0\end{split} \] are also investigated.
The theory of invariants for second-order ordinary differential equations is applied to the study of nonlinear dynamical systems dependent on a set of parameters.
For the entire collection see [Zbl 1050.35003].
34A26 Geometric methods in ordinary differential equations
34C14 Symmetries, invariants of ordinary differential equations