The authors discuss a sequence of algebraically decaying rational solutions of the Boussinesq equation \[ u_{tt}+u_{xx}-(u^2)_{xx}-\frac{1}{3}u_{xx}=0,\tag{1} \] which is a soliton equation solvable by the inverse scattering method. These rational solutions are of the form \[ u_n=2\frac{\partial^2}{\partial x^2}\ln F_n(x,t),\quad n\geq1. \] They have an interesting structure and have a similar appearance to rogue-wave solutions in the sense that they have isolated ‘lumps’. Here, \(F_n(x,t)\) is a polynomial of degree \(\frac{1}{2}n(n+1)\) in \(x^2\) and \(t^2\), and satisfies a bilinear equation of Hirota type. It is noted that the polynomials \(F_n(x,t)\) have the special form \[ F_n(x,t)=(x^2+t^2)^{n(n+1)/2}+G_n(x,t), \] where \(G_n(x,t)\) is a polynomial of degree \(\frac{1}{2}(n+2)(n-1)\) in both \(x^2\) and \(t^2\). It is also noted that these polynomials have a similar structure to those that arise in the rational solutions of the focusing nonlinear Schrödinger equation, though the coefficients in the polynomials \(G_n(x,t)\) are different.
Further, the authors use the rational solutions of the Boussinesq equation (1) to derive rational solutions of the the KP equation and compared them to those obtained from rational solutions of a focusing nonlinear Schrödinger equation equation considered in [P. Dubard and V. B. Matveev, Nonlinearity 26, No. 12, R93–R125 (2013; Zbl 1286.35226)]. It is shown that the two sets of solutions are fundamentally different and both are special cases of a more general rational solution.