Dynamical D4-D8 and D3-D7 branes in supergravity

We present a class of dynamical solutions for intersecting D4-D8 and D3-D7 brane systems in ten-dimensional type IIA and IIB supergravity. We discuss if these solutions can be recovered in lower-dimensional effective theories for the warped compactification of a general p-brane system. It is found that an effective $p+1$-dimensional description is not possible in general due to the entanglement of the transverse coordinates and the $p+1$-dimensional coordinates in the metric components. For the D4-D8 brane system, the dynamical solutions reduces to a static warped ${\rm AdS_6}\times {\rm S}^4$ geometry in a certain spacetime region. For the D3-D7 brane system, we find a dynamical solution whose metric form is similar to that of a D3-brane solution. The main difference is the existence of a nontrivial dilaton configuration in the D3-D7 solution. Then we discuss cosmology of these solutions. We find that they behave like a Kasner-type cosmological solution at $τ\to\infty$, while it reduces to a warped static solution at $τ\to0$, where $τ$ is the cosmic time.