Monads on Cartesian products of projective spaces

In this paper we establish the existence of monads on special Cartesian products of projective spaces. Special in the sense that we mimick monads on instanton bundles. We construct monads on ${\mathbb{P}}^1\times \cdots \times {\mathbb{P}}^1\times {\mathbb{P}}^3\times \cdots \times {\mathbb{P}}^3\times {\mathbb{P}}^5\times \cdots \times {\mathbb{P}}^5$. We proceed to prove stability of the kernel bundle associated to the monad and simplicity of the cohomology vector bundle. Lastly we establish the existence of monads on ${\mathbb{P}}^{a_1}\times \cdots \times {\mathbb{P}}^{a_n}$ where $a_1<a_2<\dots <a_n$, alternating even and odd or at least $a_i$ $0<i\le n$ is odd.