Abstract: This paper introduced a redundant factor λ-1 in constructing inverse scattering transform to ensure vanishing contribution of the integral along big semi-circle in upper half plane of complex λ as the radius tends to infinite.In order to verify the correctness of the solutions,this paper introduced a pair of Jost solutions with the same analytic properties of a 2×2 matrix,and then another pairs was introduced to be its right inverse confirmed by the Liouville theorem.Because they are both 2×2 matrices,the right inverse should be the left inverse too,furthemore,it is not difficult to show that these Jost solutions satisfy both Lax equations.As a result of compatibility condition,the soliton solutions definitely satisfy the L-L equation in the reflectionless.