An exact direct perturbation theory of the KdV equation with corrections is developed for multi-soliton case. After showing that the derivatives of the squared Jost functions with respect to x are the eigenfunctions of the linearized operator, suitable definitions of the adjoint functions and the inner product are introduced. Orthogonality relations are derved and the expansion of the unity in terms of the squared Jost functions is implied. The completeness relation of the squared Jost functions is shown by the generalized Marchenko equation. The final result indicates that in the expression of the completeness relation, the integral path is along the real axis from －∞ to ∞ but runs over near the origin, which is contrary to the Cauchy principal value appearing in previous works. This leads to the disappear once of the shelf behind the soliton due to perturbations, which was considered as a characterized effect in the previous theories.
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