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We explore dualities and solution-generating transformations in variouscontexts. Our focus is on the T-duality invariant form of supergravity known asdouble field theory, the $SL(5)$-invariant M-theory extended geometry, andmetrics dual under the fluid/gravity correspondence to an incompressibleNavier-Stokes fluid. In double field theory (DFT), a wave solution is shown toembed both the F1 string and the pp-wave. For the former, the Goldstone modedynamics reproduce the duality symmetric string introduced by Tseytlin. Weconsider solution-generating techniques in DFT in the presence of an isometry,firstly via Buscher-like transformations in the DFT string $\sigma$-model, andsecondly via the DFT equations of motion. In the $SL(5)$-invariant geometry, weprovide a chain rule derivation of the covariant equations of motion, andpresent a wave solution embedding the M2 brane. Lastly, solution-generatingtransformations for metrics with an isometry are considered in the context ofthe fluid/gravity correspondence. Our focus is on the vacuum Rindler metricdual to a codimension one Navier-Stokes fluid. In particular, when there is aradially directed Killing vector, the dual fluid is found to exhibit an energyscaling invariance valid to all orders in the hydrodynamic expansion.

Solution-generating transformations in duality-invariant theories and the fluid/gravity correspondence