Embedded Solitons and Conservation Law with $χ^{(2)}$ and $χ^{(3)}$ Nonlinear Susceptibilities

This paper studies embedded solitons that are confined to continuous spectrum, with $χ^{(2)}$ and $χ^{(3)}$ nonlinear susceptibilities. Bright and singular soliton solutions are obtained by the method of undetermined coefficients. Subsequently, the Lie symmetry analysis and mapping method retrieves additional solutions to the model such as shock waves, singular solitons, cnoidal waves, and several others. Finally, a conservation law for this model is secured through the Lie symmetry analysis.