Integration of systems of ODEs via nonlocal symmetry-like operators
The authors apply nonlocal symmetry-like operators to systems of two first- and two second-order ordinary differential equations in order to seek reduction quadratures. This is an important consideration since in the reduction of the order of ODEs by means of Lie point symmetries more than one local symmetry can be lost and turning into nonlocal symmetries. The authors develop a complex nonlocal symmetry method for the integration of first- and second-order restricted complex ordinary differential equations and the corresponding system of two first- and second-order ODEs. This is illustrated in two examples taken from the literature: one is a second-order system, and another one is a first-order Riccati system. The final section of the paper is devoted to the use of nonlocal Noether symmetry. A complex nonlocal Noether symmetry of a complex local or nonlocal Lagrangian can be used to obtain two first integrals for the corresponding system of two second-order ODEs. The paper ends with three illustrative examples of this methodology: the first one considers the free particle equations: \(f''=0\) and \(g''=0\). The second one is a general linear second-order ODE in the restricted complex domain: \(u''+a(x)u'+b(x)u=0\), while the third one is \(u''=u'{}^2/u+a(x)uu'+a'(x)u^2\).
scientific article; zbMATH DE number 5943166