The Instabilities Of Nonlinear Wave Systems

In this paper we study the instabilities of the initial value problems of nonlinear wave systems.First of all, for the single Klein-Gordon equation, we use the methods of energy and inequalities to study its initial value problems:where u=u(t,x) :R+ x RN C. We obtained the instability of the equation.In the next place, we still use the above methods to study a class of coupled Schrodinger equations: where (x,t)and (x,t) are complex and real-value functions respectively andm R ,x R3, t>0. The instability of the solution of the system is obtained.Then we continues study these two class equations' coupled together: coupled Klein-Gordon-Schrodinger equations:where (x,t) and (x,t) are complex and real-value functions respectively andm R,x R3, t>0. The instability of the solution of the system is obtained.At last, we extend the coupled Klein-Gordon-Schrodinger equations to generalized smooth function F(u,v): R+x R R , and we have. Then we obtained the instabilityof coupled Klein-Gordon-Schrodinger equations:where (x,t)and (x,t) are complex and real-value functions respectively andm R,x R3, t>0. The instability of the solution of the system is obtained.