The matrix Riemann-Hilbert problem of the Schrödinger-Hirota equation in nonlinear optics is investigated. The derivation of a Riemann-Hilbert problem from the Lax pair associated with the 2 \(\times \) 2 matrix spectral problem, which is linked to the Schrödinger-Hirota equation, is achieved through the application of spectral analysis to the Lax pair. Furthermore, by solving a special Riemann-Hilbert problem with vanishing scattering coefficients, an explicit representation of the N-soliton solution to the Schrödinger-Hirota equation is derived based on the asymptotic characteristics of the Riemann-Hilbert problem. It is noteworthy that the real and imaginary parts of the N-soliton solution exhibit two perfectly complementary N-breather solutions.
