Ferromagnetic Chain Equation Of Symmetry Reduction And Exact Solutions

It is well known that the research based on the application or mechanics, physics and other related subjects becomes a very important aspect of modern mathematics. A lot of mathematicians and physicists try to obtain the exact solutions of differential equations, especially for constructing exact solutions of nonlinear partial differential equations.This paper is organized as follows: In section 1, we provide some background material and notations of theory of symmetry reduction. In section 2, we compute the lie point transformation of the ferromagnetic spin equation without damping and external magnetic field (Z|→)(x, t)—(u(x, t),v(x, t), w(x, t)). Using the method pioneered by 01ver,we also systematically construct the one dimensional optimal system of the lie algebra which including 24 elements. In section 3, we construct the polynomial solutions of one dimensional ferromagnetic spin equation by using the invariant subspace method. And we also get an explicit solution of n dimensional radially symmetric ferromagnetic spin system. These solutions generally cannot be obtained via the classical symmetry method or the conditional symmetry method. Section 4 is the conclusion and the study in the future.