Integrability and inverse scattering transform of the modified Benjamin-Ono equation

Abstract This paper presents a Bäcklund transformation, the Lax representation, and conserved quantities for the modified Benjamin-Ono equation. The initial problem of the modified Benjamin–Ono equation on the line was studied by the inverse scattering transform method, presenting a nonlocal Riemann–Hilbert problem to reconstruct the explicit potential function. Furthermore, the exact N -soliton solutions and long–time asymptotic behavior are provided. We also graphically show that the propagation of soliton solutions is consistent with the result of large-time asymptotic forms It is shown that the mBO equation admits solutions in logarithmic form, which incorporates the features of solutions for the BO equation.