Exact self-similar wave solutions for the generalized (3Â +Â 1)-dimensional cubic-quintic nonlinear Schröinger equation with distributed coefficients

In this paper, a similarity transformation is presented to reduce the generalized (3 + 1)-dimensional cubic-quintic nonlinear Schrödinger equation with distributed coefficients to the related constant-coefficients one. Then a number of spatiotemporal self-similar wave solutions are constructed. Under the specific choice of the dispersion, cubic and quintic nonlinearities, phase modulation and the gain/loss, we investigate the dynamical behaviors of those spatiotemporal self-similar waves in an inhomogeneous optical fiber media.