The paper investigates a class of long-wave unstable lubrication model using Lie theory. A nonlinear self-adjoint classification of the considered equation is carried out. Without having to go into microscopic detailsof the physical aspects, non-trivial conservation laws are computed. Then, minimal set of Lie point symmetries of the discussed model is classified up to one-dimensional conjugacy classes which are further utilised one by one to construct the similarity variables to reduce the dimension of the considered model. After that, all possible phase trajectories are classified with respect to the parameters of the equation. Some travelling wave and kink-wave solutions are also showed and graphical representations are displayed to depict their propagation.

In this paper, a diverse range of travelling wave structures of perturbed Fokas–Lenells model (p-FLM) is obtained by using the extended $(G'/G^2)$-expansion technique. The existence of the obtained solutions is guaranteed by reporting constraint conditions. Then, the governing model is converted into the planer dynamical system with the help of Gallelian transformation. Every possible form of phase portraits is plotted for pertinent parameters, viz. $k$, $\beta$, $d_1$, $d_2$, $d_3$. We also used the Runge–Kutta fourth-order technique to extract the nonlinear periodic solutions of the considered problem and outcomes are presented graphically. Furthermore, quasiperiodic and chaotic behaviour of p-FLM is analysed for different values of parameters after deploying an external periodic force. Quasiperiodic–chaotic nature is observed for selected values of parameters $k$, $\beta$, $d_1$, $d_2$, $d_3$ by keeping the force and frequency of the perturbed dynamical system fixed. The sensitive analysis is employed on some initial value problems (IVPs). It is seen that de-sensitisation is present in the perturbed dynamical system while for the same values of parameters, the unperturbed dynamical system has a nonlinear periodic solution.