Generalized Darboux transformation and N-th order rogue wave solution of a general coupled nonlinear Schr\"{o}dinger equations

We construct a generalized Darboux transformation (GDT) of a general coupled nonlinear Schr"{o}dinger (GCNLS) system. Using GDT method we derive a recursive formula and present determinant representations for N-th order rogue wave solution of this system. Using these representations we derive first, second and third order rogue wave solutions with certain free parameters. By varying these free parameters we demonstrate the formation of triplet, triangle and hexagonal patterns of rogue waves.