Novel exact self-similar solitary waves in graded-index media with Kerr nonlinearity

A broad class of exact self-similar solitary wave solutions to the nonlinear Schro¨dinger equation (NLSE) are found by using the extended mapping deformation method. These novel waves can propagate inside planar, graded-index nonlinear waveguide amplifiers with self-focusing and self-defocusing Kerr nonlinearities, respectively. Furthermore, we analyze the features of the self-similar waves and their evolution. Our results in this paper include some in the literature [S.A. Ponomarenko, G.P. Agrawal, Phys. Rev. Lett. 97 (2006) 013901].