  In this paper we study the local solvability of nonlinear Schr\"odinger
equations of the form $$\p_t u = i {\cal L}(x) u + \vec b_1(x)\cdot \nabla_x u
+ \vec b_2(x)\cdot \nabla_x \bar u + c_1(x)u+c_2(x)\bar u +P(u,\bar u,\nabla_x
u, \nabla_x\bar u), where $x\in\mathbb R^n$, $t>0$, $\displaystyle{\cal L}(x) =
-\sum_{j,k=1}^n\p_{x_j}(a_{jk}(x)\p_{x_k})$, $A(x)=(a_{jk}(x))_{j,k=1,..,n}$ is
a real, symmetric and nondegenerate variable coefficient matrix, and $P$ is a
polynomial with no linear or constant terms. Equations of the form described in
with $A(x)$ merely invertible as opposed to positive definite arise in
connection with water wave problems, and in higher dimensions as completely
integrable models. Under appropriate assumptions on the coefficients we shall
show that the associated initial value problem is local well posed.
