N-bright–bright and N-dark–dark solitons of the coupled generalized nonlinear Schrödinger equations

•N-bright–bright soliton solution of a CGNLS system is given in Gram determinant form.•The Gram determinants shown to satisfy Hirota’s bilinear equations.•The explicit expressions of one, two and three bright–bright soliton solutions are given.•N-dark–dark soliton solution of the CGNLS system is also given.

We construct N-bright–bright and N-dark–dark soliton solutions of an integrable two coupled generalized nonlinear Schrödinger (CGNLS) equation for arbitrary values of system parameters. These solutions are more general than the reported one. While the bright–bright solitons are captured in the focusing regime of CGNLS equation, the dark–dark soliton solutions are identified in the defocusing regime. We present N-bright–bright solitons in the Gram determinant forms and prove that these determinant forms satisfy the Hirota bilinear equations.