This paper investigates the exact traveling wave solutions and the dynamical behavior of a generalized KdV–mKdV equation with higher-order nonlinear terms. By employing the dynamical systems method combined with an independent transformation, new families of periodic wave, solitary wave, and kink wave solutions are derived in explicit or implicit forms, many of which are reported for the first time. Importantly, we discover some unconventional features of these solutions, including nested orbital structure, the dual role of degenerate equilibrium points, and compensatory orbital hierarchy structure. These findings significantly enrich the understanding of the equation’s dynamical properties. The results highlight the influence of singular lines and degenerate equilibria on wave dynamics, offering new insights into singular traveling wave systems.
Zhang et al. (2026) studied this question.