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 $J\le 14$ and/or $J\le 9$ using the computer algebra software MAPLE. 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.