Purpose The purpose of this study is to investigate the integrability properties of a generalized breaking-soliton equation, derive various exact solutions and discuss their physical significance. Design/methodology/approach The integrability of the system is established by deriving the Hirota bilinear form, Bäcklund transformation, Lax pair and infinitely many conservation laws via the Hirota bilinear method and its close connection with Bell polynomial theory. In addition, the Hirota bilinear framework, combined with the quadratic test function method and hybrid test functions such as quadratic–exponential and quadratic–hyperbolic cosine forms is used to generate a broad class of exact and hybrid solutions. Findings A comprehensive set of integrability structures including the Hirota bilinear form, bilinear Bäcklund transformation, Lax pair and infinitely many conservation laws has been established for the model. Rich families of exact solutions are constructed, comprising complex multi-solitons, breather waves and breather–soliton interaction patterns, which are illustrated graphically. Fully localized rational lump solutions are generated via quadratic trial functions. Furthermore, more sophisticated hybrid wave structures, such as lump–stripe and lump–soliton interactions, are derived using quadratic–exponential and quadratic–hyperbolic ansätze, unveiling intricate nonlinear dynamics characterized by wave separation and fusion–fission processes. Originality/value In this paper, the Hirota bilinear form is derived via Bell polynomial theory, yielding results consistent with those obtained through the dependent variable transformation. A wide spectrum of exact solutions including complex solitons, lump solutions, breather–soliton interactions and other interaction patterns is systematically constructed. The physical characteristics of these intriguing solutions are vividly illustrated through two-dimensional and three-dimensional plots, contour diagrams and density profiles. Notably, all the solutions presented here are novel and, to the best of our knowledge, have not been reported previously in the literature.
Mandal et al. (Sat,) studied this question.