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Asynchronous wakeup schemes have been proposed for ad hoc networks to increase the energy efficiency of wireless communication. The basic idea is to allow a node to sleep when it is idle, and wakeup periodically to check if there are pending transmissions. In this paper, we examine the applicability of asynchronous wakeup schemes to the Mobile Ad Hoc NETworks (MANETs). We discover that, although it is desirable to have nodes with lower mobility to sleep more in reaction to the less-changing link states, in practice this is prohibited due to an unwanted tradeoff between the energy saving and in-time link discovery. All nodes in a network must stay awake frequently based on their highest possible moving speed to avoid network partition. To address this problem, we propose a new wakeup scheme, named Unilateral- (Uni-) scheme, for MANETs that allows nodes with slower moving speed to sleep more without losing the network connectivity. The Uni-scheme supports both the entity mobility and group mobility of nodes, thus has broad applicability. Theoretical analysis and simulation are conducted and show that the Uni-scheme can render significant energy saving as compared with the previous arts.