Yes, that is completely true
Just to add on to what Finwe said I will add this quote
Quote:
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...Legolas already lay motionless, his fair hands folded upon his brest, his eyes unclosed, blending living night and deep dream, as is the way with Elves...
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This is from the Chapter, "The Riders of Rohan". And this is the only clear description that I have read about how elves sleep. In addition during the entire chase of the orcs to rescue Merry and Pippin, Legolas is always the first up, and the most energetic. Even after they have been rushing for days, and hours straight, Legolas is still very energetic, and observant (of course
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But I don't know if this applies to Arwen, I guess it would. But I strangely never think of Arwen or Elrond sleeping in this manner. I guess it is because of the influence of the films.