Conservation laws, bilinear Bäcklund transformations and solitons for a nonautonomous nonlinear Schrödinger equation with external potentials

Under investigation in this paper is a nonautonomous nonlinear Schrödinger equation with external potentials, which can govern the dynamics of nonautonomous solitons in the nonlinear optical medium non-uniformly distributed in both the transverse and longitudinal directions. Based on the Lax pair, we present an infinite sequence of the conservation laws. Bilinear forms, bilinear Bäcklund transformations, one-, two- and N-soliton solutions under a known variable-coefficient constraint are generated via the Hirota method. With G(t)=0 and R(t)B(t) being a constant, amplitude of the soliton remains unvarying during the propagation, where t is the scaled time, G(t) is the gain/loss coefficient, B(t), the group velocity dispersion coefficient, and R(t), the nonlinearity coefficient. If we set G(t) ≠ 0 or R(t)B(t) as a variable, the amplitude becomes varying. Due to the different choices of the linear oscillator potential coefficient α(t), periodic-, parabolic-, S- and V-type solitons are observed. Meanwhile, we find that α(t) has no influence on the soliton amplitude. Interaction between the two amplitude-unvarying solitons and that between the two amplitude-varying ones are displayed, respectively. The velocity of a moving soliton always keeps varying.