In this paper, the integrable n-coupled nonlinear Schrödinger equations with mixed signs of focusing- and defocusing-type nonlinearity coefficients are gauge equivalent to the equation of Schrödinger flow from \({\mathbb {R}}\) to the pseudo-Kähler manifold \(U(n,\delta )/U(1)\times U(\delta _1,\ldots ,\delta _n),\) where \(\delta _j\in \{-1,1\},j=1,2,\ldots ,n\) , \(\delta :=\#\{\delta _1=1,\ldots ,\delta _n=1\}\) denotes that the number of ones contained in \(\delta _j\,(j=1,\ldots ,n)\) and \(U(\delta _1,\ldots ,\delta _n)\) -invariant almost Hermitian structures. This gives a geometric interpretation of the mixed n-coupled nonlinear Schrödinger equations via Schrödinger flow on the pseudo-Kähler manifold \(U(n,\delta )/U(1)\times U(\delta _1,\ldots ,\delta _n)\) . Finally, we obtain explicit soliton solutions of the 1-dimensional Schrödinger flow on the pseudo-Kähler manifold \(U(2,1)/U(1)\times U(1,1).\)
