Lifshitz/Schr\"odinger D-p-branes and dynamical exponents

We extend our earlier study of special double limits of boosted' $AdS_5$ black hole solutions to include all black D$p$-branes of type II strings. We find that Lifshitz solutions can be obtained in generality, with varied dynamical exponents, by employing these limits. We then study such double limits for boosted’ D$p$-brane bubble solutions and find that the resulting non-relativistic solutions instead describe Schr"odinger like spacetimes, having varied dynamical exponents. We get a simple map between these Lifshitz & Schr"odinger solutions and a relationship between two types of dynamical exponents. We also discuss about the singularities of the Lifshitz solutions and an intriguing thermodynamic duality.