Classical Burgers’ equation is an indispensable dynamical evolution equation that is autonomously devised by Burgers and Harry Bateman in 1915 and 1948, respectively. This important model is featured through a nonlinear partial differential equation (NPDE). Furthermore, the model plays a crucial role in many areas of mathematical physics, including, for instance, fluid dynamics, traffic flow, nonlinear acoustics, turbulence phenomena, and linking convection and diffusion processes to state a few. Thus, in the present study, an efficient analytical approach by the name ‘generalized Riccati equation approach’ is adopted to securitize the class of (2+1)-dimensional Burgers’ equations by revealing yet another set of analytical structures to the governing single and vector-coupled Burgers’ equations. In fact, the besieged method of the solution has been proven to divulge various sets of hyperbolic, periodic, and other forms of exact solutions. Moreover, the method first begins by transforming the targeted NPDE to a nonlinear ordinary differential equation (NODE), and subsequently to a set of an algebraic system of equations; where the algebraic system is then solved simultaneously to obtain the solution possibilities. Lastly, certain graphical illustrations in 2- and 3-dimensional plots are set to be depicted - featuring the evolutional nature of the resulting structures, and thereafter, analyze the influence of the Reynolds number Ra on the respective wave profiles.

The current manuscript displays elegant numerical results for cubic-quartic optical solitons associated with the perturbed Fokas–Lenells equations. To do so, we devise a generalized iterative method for the model using the improved Adomian decomposition method (ADM) and further seek validation from certain well-known results in the literature. As proven, the proposed scheme is efficient and possess a high level of accuracy.

In this article, the collective variable method to study two types of the Chen–Lee–Liu (CLL) equations, is employed. The CLL equation, which is also the second member of the derivative nonlinear Schrödinger equations, is known to have vast applications in optical fibers, in particular. More specifically, a consideration to the classical Chen–Lee–Liu (CCLL) and the perturbed Chen–Lee–Liu (PCLL) equations, is made. Certain graphical illustrations of the simulated numerical results that depict the pulse interactions in terms of the soliton parameters are provided. Also, the influential parameters in each model that characterize the evolution of pulse propagation in the media, are identified.

In this paper, computational algorithm with the aid of Mathematica software is specifically designed for the gravitational two–body problem. Mathematical module is established to find the position and velocity vectors. Application of this module for different kind of orbits (elliptic, parabolic and hyperbolic) leads to accurate results, which proved module efficiency and to be skillful. The classical power series method is to be utilized as the methodology.