Two classes of Benjamin–Ono-type equations with the Hilbert operator related to the Calogero–Moser system and the classical orthogonal polynomials

Abstract

This paper investigates two distinct classes of Benjamin–Ono(BO)-type equations with the Hilbert operator. The first class consists of equations with constant coefficients, derived from linear differential equations, with a specific focus on the Mikhailov–Novikov equation and Satsuma–Mimura equation. The second class involves BO-type equations with variable coefficients linked to orthogonal polynomials, including Hermite, Jacobi, and Laguerre polynomials. A key aspect of transforming these differential equations into BO-type equations is that the zeros of the polynomial or periodic solutions must lie in the upper half-plane. For linear and quadratic polynomials, we directly analyze their zeros to determine the solutions of corresponding BO-type equations. For higher-order polynomials, we use the pole expansion method to derive the governing many-body systems of the zeros. This study deepens our understanding of the relationship between the zeros of polynomials and the solutions of BO-type equations.