Generalized Darboux transformation and Nth order rogue wave solution of a general coupled nonlinear Schrِdinger equations

We construct a generalized Darboux transformation (GDT) of a general coupled nonlinear Schrِdinger (GCNLS) system. Using GDT method we derive a recursive formula and present determinant representations for Nth 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.