The Dirac-Schrödinger equation

Question

Let $X$ be a spin manifold and $Y$ a riemannian manifold. Let $\psi (x,y)$ be a spinor over $X$ depending on $y\in Y$. I define the Dirac-Schrödinger equation:

$${\cal D}_x \psi =\Delta_y \psi + V(\psi)$$

where ${\cal D}_x$ is the Dirac operator over $X$, $\Delta_y$ is the Laplacian operator over $Y$, and $V$ is a potential.

Can we find solutions of the Dirac-Schrödinger equation?