Optical solitons, explicit solutions and modulation instability analysis with second-order spatio-temporal dispersion

This paper obtains the dark, bright, dark-bright or combined optical and singular solitons to the nonlinear Schrodinger equation (NLSE) with group velocity dispersion coefficient and second-order spatio-temporal dispersion coefficient, which arises in photonics and waveguide optics and in optical fibers. The integration algorithm is the sine-Gordon equation method (SGEM). Furthermore, the explicit solutions of the equation are derived by considering the power series solutions (PSS) theory and the convergence of the solutions is guaranteed. Lastly, the modulation instability analysis (MI) is studied based on the standard linear-stability analysis and the MI gain spectrum is obtained.