In this article, the novel $(G'/G)$-expansion method is successfully applied to construct the abundant travelling wave solutions to the KdV–mKdV equation with the aid of symbolic computation. This equation is one of the most popular equation in soliton physics and appear in many practical scenarios like thermal pulse, wave propagation of bound particle, etc. The method is reliable and useful, and gives more general exact travelling wave solutions than the existing methods. The solutions obtained are in the form of hyperbolic, trigonometricand rational functions including solitary, singular and periodic solutions which have many potential applications in physical science and engineering. Many of these solutions are new and some have already been constructed. Additionally, the constraint conditions, for the existence of the solutions are also listed.

The paper studies the extraction of chirped soliton to Chen–Lee–Liu equation (CLLE) with the group velocity dispersion (GVD) and self-steeping coefficients that describe pulse transmission through optical monomode fibres. The chirped bright, dark and singular optical solitons are obtained and the results show that nonlinear chirp parameters strongly vary on self-steeping, GVD and spreading effects. The constraint conditions for the existence of solitons are also derived during the derivation. The results are helpful and important for understanding the propagation of optical pulses.