Self-similar localized pulses for the nonlinear Schrodinger equation with distributed cubic-quintic nonlinearity

We obtain the exact analytical self-similar bright-, dark-, and kink-solitary-wave solutions of the nonlinear Schrodinger equation with localized inhomogeneous cubic-quintic nonlinearity by employing a similarity transformation. This equation could be a model equation of stable pulse propagation beyond ultrashort range in optical fiber communication systems in inhomogeneous media. We have investigated that the self-similar bright- and dark-solitary-wave solitons show interesting compression and amplification features that can be controlled by suitably changing the phase modulation parameter and consequently varying the nonlinear parameter. We unearth a surprising connection between optical self-similar localized waves in graded-index inhomogeneous media and solitons in homogeneous media with the same type of cubic-quintic nonlinearity. Finally, the stability of the solutions is discussed numerically under finite initial perturbations.