The perturbed Korteweg–de Vries equation considered anew

Abstract

The perturbed Korteweg–de Vries equation is studied in a new way by a Green’s function formalism without use of inverse scattering methods. The Green’s function is determined by employing the Bäcklund transformation and Green’s theorem. After a thorough analysis of the exact first-order solution with regard to secular terms, a two-time scale expansion leads to the adiabatic approximation and the first-order correction, in accordance with the results of Karpman and Maslov. Contrary to statements in the literature, the term tanh2 z in the expression for the modified phase of the perturbed soliton arises as a consequence of the systematically conducted first-order perturbation theory.