Chirped travelling and localized wave solutions in the cubic-quintic nonlinear Schrödinger equation with self-steepening and self-frequency shift

A generalized nonlinear Schrödinger equation possessing cubic-quintic nonlinear components can be used to simulate how ultra-short and femtosecond optical pulses propagate through a nonlinear medium with self-frequency shift as well as self-steepening effects. Starting with an extended auxiliary equation technique, we find an elliptic differential condition with a fifth-degree nonlinear component that depicts the development of the wave amplitude in the metamaterials (MMs) by including an intensity-dependent nonlinear chirp anstaz. As a limiting example of the Jacobi elliptic function solutions for the model under consideration and taking into account the self-frequency shift as well as self-steepening effects, we present a highly rich variety of exact chirped solutions, in particular the solitary wave solutions and periodic solutions. The related chirp is governed by the parameters for self-steepening and self-frequency shift and is proportional to the intensity of the field, according to the results of the generalized nonlinear Schrödinger equation. Parametric conditions for the presence of the traveling wave structures as well as the nonlinear chirp associated with each of these solutions are additionally introduced. PACS numbers: 42.65Tg, 05.45.Yv.