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The local and non-local vector Non-linear Schrodinger Equation (NLSE) with ageneral cubic non-linearity are considered in presence of a linear termcharacterized, in general, by a non-hermitian matrix which under certaincondition incorporates balanced loss and gain and a linear coupling between thecomplex fields of the governing non-linear equations. It is shown that thesystems posses a Lax pair and an infinite number of conserved quantities andhence integrable. Apart from the particular form of the local and non-localreductions, the systems are integrable when the matrix representing the linearterm is pseudo hermitian with respect to the hermitian matrix comprising thegeneric cubic non-linearity. The inverse scattering transformation method isemployed to find exact soliton solutions for both the local and non-localcases. The presence of the linear term restricts the possible form of thenorming constants and hence the polarization vector. It is shown that forintegrable vector NLSE with a linear term, characterized by a pseudo-hermitianmatrix, the inverse scattering transformation selects a particular class ofsolutions of the corresponding vector NLSE without the linear term and map itto the solution of the integrable vector NLSE with the linear term via a pseudounitary transformation, for both the local and non-local cases.

Integrable Local and Non-local Vector Non-linear Schrodinger Equation with Balanced loss and Gain