Dark and singular optical solutions with dual-mode nonlinear Schrödinger's equation and Kerr-law nonlinearity

Recently, a new family of nonlinear physical models under the name of dual-mode nonlinear equations has been arisen. Several real-valued dual-mode equations associated with well-known single-mode equations such as KdV, mKdV, and Burgers’ have been established and studied. In this study, we provide the first attempt to establish a complex-valued dual-mode equation that corresponds to the single-mode nonlinear Shrödinger equation. Accordingly, necessary conditions on the nonlinearity and dispersion parameters for the existence of two right-left (R-L) moving wave solutions are explored. Finally, graphical analysis is provided to validate the findings of this work.