Classification of Exact Solutions to the Kaup–Boussinesq Equation and the Dynamical Patterns of Rational Solutions

Abstract. This work investigates the classification of exact solutions to the Kaup–Boussinesq equation by Darboux transformation and studies the dynamical patterns of the rational solutions by constructing a new polynomial hierarchy. The first-order and high-order Darboux transformations are proposed through solving the linear algebraic equations. It is shown that the Kaup–Boussinesq equation allows two types of single soliton solutions with the same velocity: one is the bell type bright soliton and the other is the anti-bell type dark soliton, in which the amplitudes of both types of single solitons are solely dependent on their velocities. Different forms of linear algebraic equations determine different types of Darboux transformations that can describe different types of interactions of solitons. Examples of global interaction solutions, singular interaction solutions, singular breather solutions, and rational solutions are given by choosing different seed solutions and free parameters. Additionally, it is found that the dynamical patterns of a class of singular rational solutions can be described by the zeros of a newly defined polynomial hierarchy [Formula: see text]. When the degree of the polynomial is small, this polynomial hierarchy is equivalent to the Yablonskii–Vorob’ev polynomial hierarchy. However, when the degree of the polynomial is large, they are no longer equivalent since the newly defined polynomial hierarchy has higher multiplicity at zero point, while the orders of zeros of the Yablonskii–Vorob’ev polynomial hierarchy [Formula: see text] are bounded for fixed [Formula: see text]. Moreover, it is found that the newly defined polynomial hierarchy is included in the Adler–Moser polynomial hierarchy. This offers a significant refinement of the Adler–Moser polynomial hierarchy and provides a new way to find novel wave patterns of rational solutions.