Solution of ordinary differential equations with a large Lie symmetry group
The search for solutions to ordinary differential equations by pure algebraic means (i.e., without involving quadratures) goes back to the founder of the theory, S. Lie, when he determined the integrals of ODEs of order \(n\) via determinants of order \(n+1\) associated to the point symmetries of the equation. In order to simplify the expression of the obtained integrals it is convenient to combine the method with canonical Lie-Bäcklund symmetries [\textit{N. Kh. Ibragimov}, Transformation group in mathematical physics. Moskva: Nauka (1983; Zbl 0529.53014)]. This also provides an effective procedure to study systems of ODEs. For equations admitting a small parameter, a method of approximate Lie-Bäcklund symmetries was developed by \textit{V. A. Baikov, R. K. Gazizov} and \textit{N. Kh. Ibragimov} [J. Sov. Math. 55, 1450-1490 (1991); translation from Itogi Nauki Tekh., Ser. Sovrem. Probl. Mat., Novejshie Dostizh. 34, 85-147 (1989; Zbl 0759.35003)]. Since approximate symmetries are often more numerous than classical ones, this gives a more ample margin to search for exact solutions. Examples of this have recently been developed by the author for nonlinear convection-diffusion equations [Approximate Lie group analysis and solutions of 2D nonlinear diffusion-convection equations, J. Phys. A, Math. Gen. 36, 753-764 (2003)]. In this paper, the author analyzes parametrized systems \[ y_{j}^{(n)}\approx F_{j}(x,y_{1},\dots,y_{m},y_{1}^{(n-1)},\dots,y_{m}^{(n-1)},\varepsilon), \quad 1\leq j\leq m,\tag{1} \] of \(n\)th-order ordinary differential equations,and an expression of its integrals using algebraic operations on the coordinates of \((mn+1)\) Lie-Bäcklund symmetries is found. If system (1) is invariant under \((mn+1)\) approximate Lie-Bäcklund symmetries, a determinantal criterion based on what is defined as essential integrals is proved. Slight modifications allow one to apply the method to systems of equations having different orders, and, in particular, to exact equations for which \(\epsilon=0\) holds. This is illustrated with some examples. Taking into account an additional constraint on a system of type (1), a further determinantal criterion based on the invariance of (1), under \(mn\) approximate Lie-Bäcklund symmetries, is obtained.
scientific article; zbMATH DE number 1906544