An Alternative Form of Inverse Scattering Transform for the Korteweg-de Vries Equation

In the viewpoint of physics, for the Korteweg-de Vries (KdV) equation, we care the evolution effects of soliton with respect to space in perturbation theory, so the new boundary condition u → 0 as t → ∞ is under our consideration. We derive Zakharov-Shabat Equation of inverse scattering for the KdV equation under new condition starting from the second Lax equation. Finally the space dependences of scattering data are determined corresponding to time dependences of scattering data in traditional case, which are bases for developing perturbation theory under new condition.