Abstract
We introduce a novel class of derivative nonlinear Schrödinger equations incorporating a pure derivative nonlinearity term of arbitrary order. This new model can be used as a basis for the description of femtosecond pulse propagation in highly nonlinear optical fibers beyond the Kerr limit. We demonstrate that this nonlinear wave equation offers a very rich model that supports envelope soliton solutions of different waveforms and shapes. A new form of phase structure that shows nontrivial chirping which is a nonlinear function of the wave intensity, characterizes these solutions. The reported solutions are necessarily of the soliton-type containing rational, regular (hyperbolic-secant) and soliton-like solutions of the bright-type as well as kink-type envelopes. The formation conditions for the existence of these structures are also reported.