Nayasu is a verb and its noun is nayashi. You hear these words a lot when you learn No. 3 in All Japan Kendo Kata (kata). So what does it mean?
If you do not know kata No. 3, here is a video you can check out.
Which part of kata No. 3 involves nayashi?
There is a very first tsuki (thrust) executed by one guy in kata No. 3 and the second guy steps back and avoids the thrust.
The movement the second guy is doing with his sword to avoid the thrust is "nayasu".
Then what is he doing? If you ask a Japanese what it means, they may not be able to give you an answer. If they do kendo, maybe they know.
"To nayasu" means "to soften" or "to kill/weaken the force".
So if your opponent has a really strong energy coming at you in shiai and you cannot take it and you lose your fighting spirit, the state of your mind is being "nayasu-ed".
Quick Japanese lesson: though the word "nayasu" is used in kendo, there is a more common word in Japanese for the same concept, which is "naeru".
"To naeru" is an intransitive verb so it does not take an object. It is used, in such a case of this example, like "my mind is naeta by his strong energy." "Naeta" is a past tense of "naeru".
Ok, enough grammar.
So if you nayasu your opponent's sword, it means you weaken the force of the opponent's sword Thus, the movement of kata No. 3.
If you look how he deals with the tsuki in the video, he is not just pushing down the opponent's sword. He stretches his arms and twists his own sword to the right and pulls his sword back.
You have to watch carefully on the video to see the twist. The twist is a rather small movement, but it is an effective "nayashi."
If your sensei or someone tells you to "nayasu", this is what you do. And also you hear this word almost only in kata No. 3.
However, it is very important to know what you should be doing when you hear nayasu. Remember, it does not mean "to push down."