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.

In the given research, our main goal is to explore newsoliton solutions in (2+1)-dimensional Heisenberg ferromagnetic spin chain (HFSC) equation. This was achieved via the extended (G'/G2)-expansion method. For this, we first analyse the given (2 + 1)-dimensional HFSC system of a partial differential equation (PDE) that is obtained by separating the equation into real and imaginary parts. The solution of our equation is determined as rapidly convergent sequences effectively computed by using Mathematica software. Furthermore, a special class of nonlinear wave solutions like periodic, straight, decreasing, dark-singular soliton and dark bright singular soliton were obtained in Heisenberg ferromagnetic dynamics. The amplitude of the soliton is determined by giving different values of parameters. With a suitable choice of parameters, 3D and 2D graphical illustrations are reported. The methodology applied is effective for obtaining exact solutions for various fractional partial differential equations in complex medium. The numerical module of the results obtained is studied, with fascinating figures showing the physical significance of the solutions.