Breather solutions to the nonlinear Schrödinger equation with variable coefficients and a linear potential

By using the self-similar method for obtaining localized solutions of nonlinear evolution partial differential equations, we found analytical breather solutions to the nonlinear Schrödinger equation with longitudinally variable coefficients and an arbitrary transversely linear potential. The Ma and the second-order breather solutions are derived by choosing the parameters appropriately. We discuss the influence of different parameters on the characteristics of the solutions found. We demonstrate that the parameters not only control the propagation direction of the breather, but also influence its shape and the period.