Riccati parameterized self-similar waves in tapered graded-index waveguides

We present a large family of self-similar waves by tailoring the tapering function, through Riccati parameter, in a tapered graded-index nonlinear waveguide amplifier. We show the existence of bright similaritons, self-similar Akhmediev breathers and self-similar rogue waves for generalized nonlinear Schrödinger equation with constant dispersion and nonlinearity, and a distributed gain. We illustrate the procedure to amplify the intensity of self-similar waves using isospectral Hamiltonian approach. This approach provides a handle to find analytically a wide class of tapering function and thus enabling one to control the self-similar wave structure and dynamical behavior.