Algebraic structure of space and field

Abstract

We investigate an algebraic structure of the space of solutions of autonomous nonlinear differential equations of certain type. It is shown that for these equations infinitely many binary algebraic laws of addition of solutions exist. We extract commutative and conjugate commutative groups which lead to the conjugate differential equations. Besides one is being able to write down particular form of extended Fourier series for these equations. It is shown that in space with a moving field, there always exist metrics geodesics of which are the solutions of a given differential equation and its conjugate equation. Connection between the invariant group and algebraic structure of solution space has also been studied.