Riemann–Hilbert problem for the focusing Hirota equation with counterpropagating flows

Abstract

. The focusing Hirota equation is analyzed with a general initial condition via the inverse scattering transform, whose asymptotic behavior at infinity consists of counterpropagating waves. According to some necessary conditions, including jump condition, normalization condition, residue conditions and suitable growth condition near the branch points, the inverse problem is transformed into a matrix Riemann-Hilbert (RH) problem jumping along the branch cuts and real axis, the problem is transformed into a set of linear algebraic integral equations, and the reconstruction formula of potential is successfully obtained. In addition, the zero point of the analytical scattering coefficient on the continuous spectrum is placed on a sufficiently large circle, so a modified piecewise analytical RH problem is further successfully constructed. Finally, the exact expressions of soliton solution and breathing solution of focusing Hirota equation under degenerate initial value conditions are discussed.