Exact soliton solutions and their classifications for the generalized (2+1)-dimensional Hirota–Maccari equation using the direct integral method

This study addresses the generalized (2+1)-dimensional Hirota–Maccari system, a fundamental model governing bidirectional wave-mean flow interactions in nonlinear optics, hydrodynamics, and plasma physics. Existing solution methods suffer from structural incompleteness and parameter sensitivity. We systematically derive all exact traveling wave solutions using the direct integral method integrated with the complete discrimination system for polynomials, eliminating preset functional assumptions. Under the essential parameter constraint, 18 distinct solution families are rigorously classified via discriminant criteria, including rational, hyperbolic, trigonometric, and doubly periodic Jacobi elliptic function solutions. Graphical simulations validate dynamical behaviors across criticality, pattern formation, and coherence. This framework provides a universal, parameter-classified tool for modeling femtosecond optical solitons, plasma waves, and hydrodynamic extremes, with direct experimental verifiability through amplitude–velocity correlations.