Asymptotic well-posedness for the problem of soliton perturbation

Perturbations of the KdV and MKdV equation, of the sine-Gordon equation, of a nonlinear Schrodinger equation, of the modified KdV equation, and of the Phi-4 equation are studied under the assumption that the nonperturbed problem has a one-soliton solution. It is proved that the parameters of the perturbed solution are defined uniquely and do not depend on the ambiguities that remain in the corrections if these corrections do not contain secular terms.