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The present research framework looks over complete sorted symmetry group classification and optimal subalgebras of (2+1)-dimensional modified Bogoyavlenskii–Schiff(mBSchiff) equation. It’s highly nonlinear and exhibits wave propagation in thermal pulse, sound wave, and bound particle. Using the invariance property of Lie groups, adequate infinitesimal symmetry of Lie algebra has been set up for the mBSchiff equation. A rigorous and systematized algorithm is carried out to obtain one optimal system based on the invariance feature of adjoint transformation. Further, symmetry reduction of the mBSchiff equation has been made to derive a system of ordinary differential equations with newly established similarity variables. The complete set of group invariant solutions for each corresponding subalgebras has been made. The derived solutions have diverse physical phenomena, which MATLAB simulation can quickly analyze. Thus, solutions presented here are kink, positon, soliton, doubly soliton, negaton, multisoliton types, which add on some meaningful physical aspects of the research.

In this paper, we introduced and established some group invariant results of [Formula: see text]-dimensional mKdV–Calogero–Bogoyavlenskii–Schiff equation. Using the one-parameter Lie-group of transformations, we explored various closed-form solutions. The procedure minimizes the number of independent variables by one in every proceeding stage leading to form a system of the ordinary differential equations. The nature of solutions is investigated both analytically and physically through their evolutionary profiles by considering adequate choices of arbitrary functions and constants. The obtained results have been plotted with the aid of numerical simulation to obtain a significant appearance of the traced results. Simulation is carried out by taking an adequate option of arbitrary constants and functions, applying MATLAB code through progressing profiles. Wave solutions derived here are positons, multiple solitons, negaton and kink types which are shown through graph analysis.

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The objective of the present article is to seek new explicit solutions to the (2+1)-dimensional Nizhnik-Novikov-Veselov(NNV) equation. The NNV system is highly nonlinear in nature and is a known isotropic Lax extension of the Korteweg-de Vries model. The similarity transformation method has been used to systematically reduce the NNV equation into ordinary differential equations(ODEs). The new exact solutions have been obtained by solving the obtained ODEs based on the formed relationships. The resulting soliton solutions contain some arbitrary constants and functions. The use of appropriate functions and constants highlighted that the solutions of the NNV equation might be soliton, multisoliton, parabolic, doubly soliton and trigonometric. Moreover, the stability of the corresponding dynamical system has been investigated using bifurcation theory with different parametric regions. The figures obtained during MATLAB simulation supported the dynamic features of the derived solutions. Finally, we explore incredible aspects of the exact wave solutions via phase portraits. The phase portraits validate the existence of some families of homoclinic and periodic orbits about the equilibrium points, respectively.