Research On Static Solutions And Blow Up Problem Of Schr(?)dinger Map

The Landau-Lifshitz equation is a very important equation for describing ferromagnetic fluid.The Schr?dinger map is a very important part of the equation.And it is the most difficult part of the equation,which plays a decisive role in the many properties of the equation.Because ferromagnetic fluids are ubiquitous in life,many people in the world are studying this equation.This paper is mainly divided into two parts to study the equation.In the first part,we construct the static solution of the Schr?dinger map with anisotropic fields.In the second part,we analyze the blow up problem of the two equivariant initial values of the SM equation.The first chapter mainly introduces the physical background,the history of the development of the Schr?dinger map equation and the main work of this paper.At present,the conclusion of the Schr?dinger map equation with anisotropic field is very few,so we first study the static solution.In the second chapter,by constructing some special transformations,we find the relation between the equation and the sine-Gordon equation.This relation is very important,which shows that all the solutions of sine-Gordon equations can be transformed into Schr?dinger map by a transformation.In the paper,a large class of static solutions is constructed through the Hirota method and properties of such solutions are proved.So we can show the properties of the magnetization motion described by Schr?dinger map equation,and then extend from two-dimensional case to high-dimensional case.Finally,the dynamic solution of the static perturbed equation is constructed.In physics,the blow up solution of Schr?dinger map equation is very significant,but the blasting problem of the equation has never been completely solved.In the third chapter,we mainly discuss the explosion problem of two equivariant initial value.First we get the equivalent equation through the transformation of Frenet frame and the decomposition of linear Hamilton operator.Then the error equation of the approximate solution is obtained through the iteration of the parameters,and the equation can be treated by some existing means.The way to deal with this problem is to prove the existence of blow up solution for the error equation first,and then by the estimation of the scattering energy and the stability of Lyapunov function control,we can get that the original equation exist blow up solution.