Abstract
The theme of this piece of research is to investigate the collective variable (CV) as well as semi-inverse techniques to explore a significant model of cold bosonic atoms in a zig-zag optical lattice. The system is reduced to an important equation by utilizing the continuum approximation and explains the soliton’s dynamics in sense of pulse variables. These parameters are amplitude, temporal position, chirp, width, frequency, and phase which are termed as collective variables (CVs). The proposed methods are more straightforward, succinct, accurate, and simple to calculate. Furthermore, to employ the computational counterfeit on the system of six ordinary differential equations that denote all the CVs incorporated in the supposed ansatz, a well-established computational method which is the Runge–Kutta scheme of order four is applied. The CV method is exerted to resolve the evolution of pulse parameters with the propagation distance and graphical illustrations which are also given. Moreover, figures reveal the fascinating periodic oscillations of frequency, width, amplitude, and chirp of soliton. Solitons and their numerical behavior to interpret fluctuations in CVs are presented for several values of super-Gaussian pulse parameters. Also, the results for the semi-inverse method which are bright solitons are provided in the form of 2D, 3D, and density plots for the distinct values of the fractal parameter to understand their physical significance. This scheme is effective in finding variational principles of various nonlinear evolution equations. Some compelling characteristics pertaining to the current scrutiny are also deduced.