Butterfly-like waves in the nonlinear Schrödinger equation with a combined dispersion term

All possible optical waves in the nonlinear Schrödinger equation with a combined dispersion term are determined according to different parameters’ regions. First, we find this equation has some popular exotic solutions, such as peaked waves, looped and cusped waves. What is more interesting, this equation admits some very particular waves, such as double kinked waves and butterfly-like waves. These new two types of solutions have not been reported in the literature regarding the study of other nonlinear equations.

► Our method is an improvement of others’ work. Our method can deal with more complicated equations.
► We obtain very detailed classification of optical waves in the nonlinear Schrödinger equation.
► We obtain some new solutions such as butterfly-like waves, which have not been reported.