ABSTRACT
In this chapter, the authors aim to prove that the inverse scattering problem (ISP) for the perturbed string equation in characteristic variables on the whole axis is associated with the two-dimensional generalization from the one-dimensional Korteweg-de Vries (KdV) equation. They describe the direct and ISP for this equation. The ISP for the perturbed string equation in characteristic variables on the whole axis is studied. Using the generalized Lax equation generated by the perturbed string equation, the authors also aim to derive the time-evolution of the scattering operator and the two-dimensional generalization from the one-dimensional Korteweg-de Vries equation. They discuss the two-dimensional generalization from the KdV equation from the generalized Lax equation, which is generated by the associated perturbed string equation. With the help of the generalized Lax equation, scattering problem is established.
