The authors consider the EM-generalization of the classical KdV equation: \(u_t -6uu_x +u_{xxx}=0\) for which it has been established that, if the evolution variable \(u(x,t)\) is viewed as a stationary quantum potential at \(t=0\), then a one-dimensional relativistic quantum model can be associated. That quantum model is represented by the Schrödinger equation.
In this article, the authors derive a new generalized form of the KdV equation with variable coefficients having damped and forced terms through effective mass (EM) Lax pair formulation in a position dependent mass (PDM) framework. In the constant mass (CM) limit \(m \to 1\), the equation reduces to the classical form above. They obtain an infinite number of conserved quantities for the generated equation and prove that it is an integrable system. The inverse scattering transform method is applied to obtain the general solution of the EM-generalized KdV and provide an explicit form of the \(N\)-soliton solution for reflectionless potentials. By choosing a linear time evolution rule for the mass function, the authors obtain mass deformed soliton solutions which display distinct properties from the classical KdV soliton solutions.