Nonlinear separable equations in linear spaces and commutative Leibniz algebras
We consider nonlinear equations in linear spaces and algebras which can be solved by a “separation of variables” obtained due to Algebraic Analysis. It is shown that the structures of linear spaces and commutative algebras (even if they are Leibniz algebras) are not rich enough for our purposes. Therefore, in order to generalize the method used for separable ordinary differential equations, we have to assume that in algebras under consideration there exist logarithmic mappings. Section 1 contains some basic notions and results of Algebraic Analysis. In Section 2 we consider equations in linear spaces. Section 3 contains results for commutative Leibniz algebras. In Section 4 basic notions and facts concerning logarithmic and antilogarithmic mappings are collected. Section 5 is devoted to separable nonlinear equations in commutative Leibniz algebras with logarithms.