Complex soliton dynamics in the M-truncated fractional nonlinear long-wave system: lump, breather and M-shape structures

In this study, we investigate the fractional modified regularized long-wave Burger (fMRLW–Burger) equation, a governing model widely used to explain the evolution of nonlinear surface water waves. To more accurately represent memory-dependent and nonlocal transport effects, the M-truncated fractional derivative is incorporated, providing enhanced flexibility in the dynamical formulation. A hybrid analytical framework is developed, placing particular emphasis on the Hirota bilinear method, where multiple trial functions are used to systematically derive exact wave structures. Complementing this, the extended direct algebraic method (EDAM) is established to further enrich the class of solutions. Through this combined strategy, we derive a diverse family of exact solutions, including rational profiles, mixed trigonometric–hyperbolic forms, dark and periodic waves, as well as composite structures such as dark–bright and bright–dark solitons. Furthermore, the Hirota-based construction allows the formulation of lump-type wave solutions, revealing highly localized phenomena such as rogue-wave pulses and breather-like patterns. The obtained solution families reveal previously unreported composite wave structures and explicit lump-type configurations under the M-truncated fractional operator. The analysis demonstrates that the fractional parameters significantly influence wave amplitude, localization, and stability characteristics, providing enhanced control over nonlinear wave modulation. These findings confirm the effectiveness of the hybrid Hirota–EDAM framework in capturing complex dispersive–dissipative interactions within fractional long-wave dynamics. Sensitivity analysis is conducted to reveal how parameter variations influence the system’s dynamical response, offering deeper insight into wave modulation and control. All analytical outcomes are visually illustrated through 2D profiles, 3D surfaces, and contour projections using symbolic computation. Thus, the results enhance the theoretical understanding of nonlinear wave propagation and highlight the effectiveness of Hirota’s bilinear approach in uncovering complex wave interactions within fluid dynamics and related fields.