Higher-order b-family systems: Abundant pseudo-peakons, peakons, and a conjecture

We investigate the fifth-, seventh-, and ninth-order b-family systems and construct their pseudo-peakon and peakon solutions. The fifth-order system admits only solutions (both pseudo-peakons and peakons) that are independent of the system parameter b, whereas the seventh- and ninth-order systems support pseudo-peakons that are independent of b and peakons that may be either independent of or explicitly dependent on b. Pseudo-peakon solutions contain arbitrary constants whose number increases with system order, which yields a diverse range of profiles (M-shaped, W-shaped, double-peakon, and soliton-like) and permits parameter-driven transformations from lower- to higher-order pseudo-peakons. Conversely, under appropriate constraints, it is possible to recover pseudo-peakon solutions from peakons. The absence of b-dependent solutions in the fifth-order case highlights a qualitative distinction between lower- and higher-order members of the family and emphasizes the role of higher-order terms in enabling parameter-dependent coherent structures. Motivated by these results, we conjecture that a general (2n+1)-order b-family system admits a third-order pseudo-peakon of the proposed form; this conjecture has been verified for the fifth-, seventh-, ninth-, and eleventh-order cases. Together, these findings clarify how the parameter b and increased differential order shape the existence and diversity of coherent solutions in the b-family hierarchy.