Consider the perturbed KdV Equation $$u_t-6uu_x+u_{xxx}=\epsilon u$$,I want to use perturbative expansion to construct the solution as the form $$u=u(x,t;\epsilon)=\sum_{n=0}^\infty\epsilon^n u_n(x,t)$$ The following is my question:

1.Does there exist the the solution in that form?How to prove it is convergent to the exact solution?

2.If so, we have $$u_{0t}-6u_0u_{0x}+u_{0xxx}=0$$it is the KdV equation,which can be solved by the inverse scattering method.And $$u_{1t}+6(u_0u_{1x}+u_1u_{0x})+u_{1xxx}=u_0$$,can anyone help me to prove there exists the solution of $u_1$ from this equation?