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In the present work we explore the potential of models of the discretenonlinear Schr\"odinger (DNLS) type to support spatially localized andtemporally quasiperiodic solutions on top of a finite background. Suchsolutions are rigorously shown to exist in the vicinity of the anti-continuum,vanishing coupling limit of the model. We then use numerical continuation toillustrate their persistence for finite coupling, as well as to explore theirspectral stability. We obtain an intricate bifurcation diagram showing aprogression of such solutions from simpler ones bearing single- and two-siteexcitations to more complex, multi-site ones with a direct connection of thebranches of the self-focusing and self-defocusing nonlinear regime. We furtherprobe the variation of the solutions obtained towards the limit of vanishingfrequency for both signs of the nonlinearity. Our analysis is complemented byexploring the dynamics of the solutions via direct numerical simulations.

Existence, stability and spatio-temporal dynamics of time-quasiperiodic solutions on a finite background in discrete nonlinear Schr\"odinger models