The Negative Flow of the Benjamin–Ono Equation

ABSTRACT This paper introduces and investigates the negative flow of the Benjamin–Ono (BO) equation, an integrable integrodifferential model that had not previously been studied. We establish its bilinear form and derive both a bilinear Bäcklund transformation and a nonlinear superposition formula. These structures are then combined to develop a unified solution approach, which systematically generates exact solutions in determinant form. The strength of this approach lies not only in its compact algebraic representation but also in its adaptability to rigorous analysis involving the Hilbert transform, thereby revealing deeper structural properties of the system. Within this framework, we obtain a Lax pair and explicitly construct two fundamental classes of solutions—multisoliton solutions and multiphase solutions. This study not only enriches the theory of the BO hierarchy but also provides a theoretical foundation for the investigation of broader classes of integrable integrodifferential equations.