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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?

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There are techniques other than the inverse scattering method to solve the KdV equation: energy estimates, semigroup theory etc. These techniques can be used to prove differentiability with respect to the parameter $\epsilon$. Since $\epsilon$ can be complex, this can be used to justify power series expansions.

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This is covered in Ablowitz and Segur (Book) for soliton initial conditions. An alternative method is given in a paper by Allen Newel. Neither of which use inverse scattering. Here multiples scales is used and much of the interest was on tails which would trail behind the soliton. Although, this doesn't exactly answer your question, since these methods are not concerned with convergence as much as accurate approximations.

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