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

Abstract

This paper presents a B{"a}cklund 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 in the inverse problem to reconstruct the explicit potential function. Further, the exact $N$-soliton solutions and long--time asymptotic behavior are provided, respectively. We also graphically show that the propagation of soliton solutions is consistent with the result of large-time asymptotic forms. As a revised version, the mBO equation incorporates the solution features of the BO equation, and its solutions are presented in logarithmic form, which seeks to describe related physical phenomena more accurately. Our results will contribute to further exploration of physical and mathematical problems related to the mBO equation.