Abstract
In this article, the propagation of breathers, multiwave, M-shape solitons, grey-black optical solitons, periodic cross rational (PCR) solitons, kink cross rational (KCR) solitons will be discussed under the effect of self-steepening and third-order of linear dispersion in single-mode fibers. All new analytical solutions are found by utilizing the symbolic computation with ansatz functions scheme and logarithmic transformation. Moreover two kinds of interaction of M-shape soliton with kink wave will be examined. We apply nine different approaches, namely, three-waves technique, homoclinic breather technique, M-shape solitons, single exponential form (one-kink), double exponential form (two-kink), PCR, KCR, grey optical soliton and black optical soliton for governing model. Moreover, stability characteristics of all solutions are evaluated.