Darboux transformation for the Zn-Hirota systems

Hirota equation is a modified nonlinear Schrodinger (NLS) equation, which takes into account higher order dispersion and delay correction of cubic nonlinearity. The propagation of the waves in the ocean is described, and the optical fiber can be regarded as a more accurate approximation than the NLS equation. Using the algebraic reductions from the Lie algebra gl(n, C) to its commutative subalgebra Z(n), we construct the general Z(n)-Hirota systems. Considering the potential applications of two-mode nonlinear waves in nonlinear optical fibers, including its Lax pairs, we use the algebraic reductions of the Lie algebra gl(2, C) to its commutative subalgebra Z(2) = C[Gamma]/[Gamma(2)]. Then, we construct Darboux transformation of the strongly coupled Hirota equation, which implies the new solutions of (q([1],) r([1])) generated from the known solution (q, r). The new solutions (q([1]), r([1])) furnish soliton solutions and breather solutions of the strongly coupled Hirota equation. Furthermore, using Taylor series expansion of the breather solutions, the rogue waves of the strongly coupled Hirota equation can be given demonstrably. It is obvious that different images can be obtained by choosing different parameters.