A weakly coupled Hirota equation and its rogue waves

The Hirota equation, a modified nonlinear Schrodinger (NLS) equation, takes into account higher-order dispersion and time-delay corrections to the cubic nonlinearity. Its wave propagation is like in the ocean and optical fibers can be viewed as an approximation which is more accurate than the NLS equation. By considering the potential application of two mode nonlinear waves in nonlinear fibers under a certain case, we use the algebraic reductions from the Lie algebra gl(2, C) to its commutative subalgebra Z(2) = C[Gamma]/(Gamma(2)) and Gamma = (delta(i,j + 1))(ij) to define a weakly coupled Hirota equation (called Frobenius Hirota equation) including its Lax pair, in this paper. Afterwards, Darboux transformation of the Frobenius Hirota equation is constructed. The Darboux transformation implies the new solutions of (q([1]), r([1])) generated from the known solution (q, r). The new solutions (q([1]), r([1])) provide soliton solutions, breather solutions of the Frobenius Hirota equation. Further, rogue waves of the Frobenius Hirota equation are given explicitly by a Taylor series expansion of the breather solutions. In particular, by choosing different parameter values for the rogue waves, we can get different images.