Novel picosecond wave solutions and soliton control for a higher-order nonlinear Schrödinger equation with variable coefficient

We try to study the propagation of ultrashort optical pulses in dispersion-decreasing fibers using the variable coefficients higher-order nonlinear Schrödinger (VCHNLS) equation. By applying the Jacobi expansion technique combined with a similarity transformation, we reduce the VCHNLS equation to a third-order nonlinear ordinary differential equation. The similarity transformation involves two variables and an exponential term related to the real gain coefficient. Integrability conditions are necessary for this reduction. Using the Jacobi elliptic expansion technique, we find various wave function solutions, including novel periodic waves and previously reported dark solitons. The new Jacobi SN wave solution and its limiting hyperbolic wave function are influenced by the real gain coefficient and the third-order dispersion variable, which control the wave’s shape, amplitude, and phase. This solution is new and gives periodic, kink, bright solitons according to different choices of the controlled variables. Moreover, we also analyze the stability of these solutions.