Soliton solutions of the nonlinear Schrödinger equation with the dual power law nonlinearity and resonant nonlinear Schrödinger equation and their modulation instability analysis

In optical fibers, the higher order nonlinear Schrödinger equations describe propagation of ultra-short pluse. The proposed modified simple equation method is employed to the nonlinear higher order Schrödinger equations for soliton solutions. The balance numbers in these Schrödinger equations are positive non-integers. The obtain solitary solutions are also presented by graphically. The modulation instability analysis shows the stability and movement of the waves, which confirms that all obtained solutions are analytical and stable. This is a new and standardized method which is applicable to solve different kind of problems in mathematics and physics.