Optical solitons and conservation laws with quadratic-cubic nonlinearity

We consider an extended nonlinear Schrödinger equation with competing quadratic-cubic nonlinearity. The model appears as an approximate form of a relatively dense quasi-one-dimensional Bose-Einstein condensate with repulsive contact interactions between atoms and a long-range dipole-dipole attraction between them. Exact analytical soliton solutions for the model are derived by utilizing the method of undetermined coefficients. The solutions comprise bright, dark, and singular solitons. Furthermore, we present a new structure which describes the evolution of two different type solitons in the form of W-shaped soliton solution and a bright soliton solution on a continuous-wave background. The constraint conditions for the existence of these solutions are also exhibited. The numerical evolution of the obtained solutions is also presented.