Lie group analysis of second order non-linear differential equations with retarded argument

Abstract

A complete classification scheme of DDE remains illusive despite of many dedicated efforts on the validity, versatility and successful implementations of this method. This paper extends the classification method of second order linear retarded delay differential equations (RDDEs) to second order non-linear RDDEs as solvable Lie algebras. The approach used avoids changing the space variables since the delay differential equations cannot possess an equivalent transformation under the change of variables. The infinitesimal generator of DDEs is used to determine the associated symmetry group. The equations are solved and the solvable Lie algebras spanned by these parameters are obtained by satisfying the inclusion property. These results and the successful implementation form the basis for the classification of non-linear delay differential equations of retarded type to solvable Lie algebra. It shows that the method is valid and feasible to the study of non-linear second order retarded delay differential equations. This classification allows us to study many natural phenomena describing by non-linear RDDEs.