Generalization of the Kaup-Newell Inverse Scattering Formulation and Darboux Transformation

In this paper the Kaup-Newell (KN) inverse scattering formulation are generalized, and a new Darboux transformation is introduced for it. Some Darboux covariant (1+1) dimensional soliton equations associated with the generalized KN formulation are derived systematically, which include not only the derivative nonlinear Schrödinger (DNLS), the modified KdV, the sine-Gordon and the massive Thirring equations but also some new nonlinear time evolution equations. Multi-soliton solutions of the DNLS equation are constructed in the determinant form using the Darboux transformation, which include some new solutions, such as quasi-periodic solutions and soliton solutions under the quasi-periodic backgrounds.