Abstract The complex coupled Maccari (CCM) system plays a remarkable role in quantum field theory, hydrodynamics, plasma physics, etc. This unified method stands upon the fact that linear and integrable nonlinear equations have the distinguishing characteristics that they possess a Lax pair formulation. Under this investigation, we have applied the unified method to the CCM system to find the ample exact travelling wave solutions and explore the various nature of wave profiles of the obtained solutions. Scores of different genre wave solutions of the CCM system are attained such as trigonometric function, rational function, hyperbolic function, exponential function, etc. with several free parameters which are dependent on the velocity of the solitary wave. Based on the values of the associated free constants, wave number, and velocity, the accomplished solitons represent multiple shapes and the nature of the wave profile is demonstrated by illustrating 3D figures through Mathematica software. Especially, 2D figures clarify the natures of the wave profile whereas the change in the velocity contacts to the value of wave number (λ)documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$lambda )$$end{document}. As a result, bell-shaped, anti-bell shapes, periodic, two solitons, bright soliton are demonstrated by the obtained solitary wave solutions. Furthermore, the obtained results ensure that the method is informative and powerful to investigate various nonlinear evolution equations that appeared in engineering, mathematics, and physics. Moreover, the graphs of wave profiles provide an important and informative clue to elaborate the stated system in further study.
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