Schrödinger equation with time-dependent mass function and associated generalized KdV equation

Abstract

We show that a time-dependent disturbance term could be embedded into a Korteweg–de Vries (KdV) equation. This is accomplished through a Lax pair formulation of a linearized scattering problem corresponding to the Schrödinger equation. A new generalized KdV equation is obtained by considering the time-dependent mass function. Choosing initial data for the stationary Schrödinger potential, we solve the direct scattering problem. Then, in the inverse scattering problem, the Gel’fand–Levitan integral equation is solved to derive the solution for our new generalized KdV equation. Finally, an elliptic functional form is chosen for the mass function to obtain an elliptic soliton solution.