Consider the perturbed KdV Equation$$u_t-6uu_x+u_{xxx}=\epsilon u,u(x,0)=f(x)$$where $f(x)=v(x,0)$，$v(x,t)$ is a soliton solution.$u$ satisfies the condition$u\to 0 $when $|x|\to\infty$
I want to use semi-group of operator theory to prove the existence of the solution of this problem.The following is what I have done.
First consider $$u_t+u_{xxx}=0,u(x,0)=f(x)$$ Use Fourier Transform we get $\overline{u}=e^{i\xi^3t}\overline{f}$,then $$u=F^{-1}(e^{i\xi^3t}\overline{f})=\int k(x-y,t)f(y)dy$$,define the operator $S(t)$ as $$S(t)(f)=\int k(x-y,t)f(y)dy$$. I want to prove this operator is a semi-group of operator.Then I want to prove the solution of the perturbed KdV Equation is $$u(x,t)=S(t)f(x)+\int_0^t S(t-s)(6u(s)u_x(s)+\epsilon u(s))ds$$,can anyone help me?
