Peakons and Pseudo-Peakons of Higher Order b-Family Equations

This paper explores the rich structure of peakon and pseudo-peakon solutions for a class of higher-order b-family equations, referred to as the J-th b-family (J-bF) equations. We propose several conjectures concerning the weak solutions of these equations, including a b-independent pseudo-peakon solution, a b-independent peakon solution, and a b-dependent peakon solution. These conjectures are analytically verified for J ≤ 14 and/or J ≤ 9 using the symbolic computation system MAPLE, which includes a built-in definition of the higher-order derivatives of the sign function. The b-independent pseudo-peakon solution is a 3rd-order pseudo-peakon for general arbitrary constants, with higher-order pseudo-peakons derived under specific parameter constraints. Additionally, we identify both b-independent and b-dependent peakon solutions, highlighting their distinct properties and the nuanced relationship between the parameters b and J. The existence of these solutions underscores the rich dynamical structure of the J-bF equations and generalizes previous results for lower-order equations. Future research directions include higher-order generalizations, rigorous proofs of the conjectures, interactions between different types of peakons and pseudo-peakons, stability analysis, and potential physical applications. These advancements significantly contribute to the understanding of peakon systems and their broader implications in mathematics and physics.