The Mixed n-Coupled Nonlinear Schrödinger Equations and Related Schrödinger Flow

Abstract 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 $$U(2,1)/U(1)\times U(1,1).$$