  In this paper, based on the nonlinear fractional equations proposed by
Ablowitz, Been, and Carr in the sense of Riesz fractional derivative, we
explore the fractional coupled Hirota equation and give its explicit form.
Unlike the previous nonlinear fractional equations, this type of nonlinear
fractional equation is integrable. Therefore, we obtain the fractional
$n$-soliton solutions of the fractional coupled Hirota equation by inverse
scattering transformation in the reflectionless case. In particular, we analyze
the one- and two-soliton solutions of the fractional coupled Hirota equation
and prove that the fractional two-soliton can also be regarded as a linear
superposition of two fractional single solitons as $|t|\to\infty$. Moreover,
under some special constraint, we also obtain the nondegenerate fractional
soliton solutions and give a simple analysis for them.
